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<H1> 25. Statistics </H1>
<!--docid::SEC176::-->
<P>

I hope that someday Octave will include more statistics functions.  If
you would like to help improve Octave in this area, please contact
<A HREF="mailto:bug@octave.org">bug@octave.org</A>.
</P>
<P>

<TABLE BORDER="0" CELLSPACING="0">
<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="octave_26.html#SEC177">25.1 Basic Statistical Functions</A></TD><TD>&nbsp;&nbsp;</TD><TD ALIGN="left" VALIGN="TOP"></TD></TR>
<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="octave_26.html#SEC178">25.2 Tests</A></TD><TD>&nbsp;&nbsp;</TD><TD ALIGN="left" VALIGN="TOP"></TD></TR>
<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="octave_26.html#SEC179">25.3 Models</A></TD><TD>&nbsp;&nbsp;</TD><TD ALIGN="left" VALIGN="TOP"></TD></TR>
<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="octave_26.html#SEC180">25.4 Distributions</A></TD><TD>&nbsp;&nbsp;</TD><TD ALIGN="left" VALIGN="TOP"></TD></TR>
</TABLE>
<P>

<A NAME="Basic Statistical Functions"></A>
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<H2> 25.1 Basic Statistical Functions </H2>
<!--docid::SEC177::-->
<P>

<A NAME="doc-mean"></A>
<A NAME="IDX775"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>mean</B> <I>(<VAR>x</VAR>, <VAR>dim</VAR>, <VAR>opt</VAR>)</I>
<DD>If <VAR>x</VAR> is a vector, compute the mean of the elements of <VAR>x</VAR>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>mean (x) = SUM_i x(i) / N
</pre></td></tr></table>If <VAR>x</VAR> is a matrix, compute the mean for each column and return them
in a row vector.
<P>

With the optional argument <VAR>opt</VAR>, the kind of mean computed can be
selected.  The following options are recognized:
</P>
<P>

</P>
<DL COMPACT>
<DT><CODE>&quot;a&quot;</CODE>
<DD>Compute the (ordinary) arithmetic mean.  This is the default.
<P>

</P>
<DT><CODE>&quot;g&quot;</CODE>
<DD>Computer the geometric mean.
<P>

</P>
<DT><CODE>&quot;h&quot;</CODE>
<DD>Compute the harmonic mean.
</DL>
<P>

If the optional argument <VAR>dim</VAR> is supplied, work along dimension
<VAR>dim</VAR>.
</P>
<P>

Both <VAR>dim</VAR> and <VAR>opt</VAR> are optional.  If both are supplied,
either may appear first.
</P>
</DL>
<P>

<A NAME="doc-median"></A>
<A NAME="IDX776"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>median</B> <I>(<VAR>x</VAR>)</I>
<DD>If <VAR>x</VAR> is a vector, compute the median value of the elements of
<VAR>x</VAR>.
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>            x(ceil(N/2)),             N odd
median(x) =
            (x(N/2) + x((N/2)+1))/2,  N even
</pre></td></tr></table>If <VAR>x</VAR> is a matrix, compute the median value for each
column and return them in a row vector.
</DL>
<P>

<A NAME="doc-std"></A>
<A NAME="IDX777"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>std</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX778"></A>
<DT><U>Function File:</U>  <B>std</B> <I>(<VAR>x</VAR>, <VAR>opt</VAR>)</I>
<DD><A NAME="IDX779"></A>
<DT><U>Function File:</U>  <B>std</B> <I>(<VAR>x</VAR>, <VAR>opt</VAR>, <VAR>dim</VAR>)</I>
<DD>If <VAR>x</VAR> is a vector, compute the standard deviation of the elements
of <VAR>x</VAR>.
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>std (x) = sqrt (sumsq (x - mean (x)) / (n - 1))
</pre></td></tr></table>If <VAR>x</VAR> is a matrix, compute the standard deviation for
each column and return them in a row vector.
<P>

The argument <VAR>opt</VAR> determines the type of normalization to use. Valid values
are
</P>
<P>

</P>
<DL COMPACT>
<DT>0:
<DD>  normalizes with N-1, provides the square root of best unbiased estimator of 
  the variance [default]
<DT>1:
<DD>  normalizes with N, this provides the square root of the second moment around 
  the mean
</DL>
<P>

The third argument <VAR>dim</VAR> determines the dimension along which the standard
deviation is calculated.
</P>
</DL>
<P>

<A NAME="doc-cov"></A>
<A NAME="IDX780"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>cov</B> <I>(<VAR>x</VAR>, <VAR>y</VAR>)</I>
<DD>If each row of <VAR>x</VAR> and <VAR>y</VAR> is an observation and each column is
a variable, the (<VAR>i</VAR>, <VAR>j</VAR>)-th entry of
<CODE>cov (<VAR>x</VAR>, <VAR>y</VAR>)</CODE> is the covariance between the <VAR>i</VAR>-th
variable in <VAR>x</VAR> and the <VAR>j</VAR>-th variable in <VAR>y</VAR>.  If called
with one argument, compute <CODE>cov (<VAR>x</VAR>, <VAR>x</VAR>)</CODE>.
</DL>
<P>

<A NAME="doc-corrcoef"></A>
<A NAME="IDX781"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>corrcoef</B> <I>(<VAR>x</VAR>, <VAR>y</VAR>)</I>
<DD>If each row of <VAR>x</VAR> and <VAR>y</VAR> is an observation and each column is
a variable, the (<VAR>i</VAR>, <VAR>j</VAR>)-th entry of
<CODE>corrcoef (<VAR>x</VAR>, <VAR>y</VAR>)</CODE> is the correlation between the
<VAR>i</VAR>-th variable in <VAR>x</VAR> and the <VAR>j</VAR>-th variable in <VAR>y</VAR>.
If called with one argument, compute <CODE>corrcoef (<VAR>x</VAR>, <VAR>x</VAR>)</CODE>.
</DL>
<P>

<A NAME="doc-kurtosis"></A>
<A NAME="IDX782"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>kurtosis</B> <I>(<VAR>x</VAR>, <VAR>dim</VAR>)</I>
<DD>If <VAR>x</VAR> is a vector of length <EM></EM>, return the kurtosis
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>kurtosis (x) = N^(-1) std(x)^(-4) sum ((x - mean(x)).^4) - 3
</pre></td></tr></table><P>

of <VAR>x</VAR>.  If <VAR>x</VAR> is a matrix, return the kurtosis over the
first non-singleton dimension. The optional argument <VAR>dim</VAR>
can be given to force the kurtosis to be given over that 
dimension.
</P>
</DL>
<P>

<A NAME="doc-mahalanobis"></A>
<A NAME="IDX783"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>mahalanobis</B> <I>(<VAR>x</VAR>, <VAR>y</VAR>)</I>
<DD>Return the Mahalanobis' D-square distance between the multivariate
samples <VAR>x</VAR> and <VAR>y</VAR>, which must have the same number of
components (columns), but may have a different number of observations
(rows).
</DL>
<P>

<A NAME="doc-skewness"></A>
<A NAME="IDX784"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>skewness</B> <I>(<VAR>x</VAR>, <VAR>dim</VAR>)</I>
<DD>If <VAR>x</VAR> is a vector of length <EM></EM>, return the skewness
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>skewness (x) = N^(-1) std(x)^(-3) sum ((x - mean(x)).^3)
</pre></td></tr></table><P>

of <VAR>x</VAR>.  If <VAR>x</VAR> is a matrix, return the skewness along the
first non-singleton dimension of the matrix. If the optional
<VAR>dim</VAR> argument is given, operate along this dimension.
</P>
</DL>
<P>

<A NAME="doc-values"></A>
<A NAME="IDX785"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>values</B> <I>(<VAR>x</VAR>)</I>
<DD>Return the different values in a column vector, arranged in ascending
order.
</DL>
<P>

<A NAME="doc-var"></A>
<A NAME="IDX786"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>var</B> <I>(<VAR>x</VAR>)</I>
<DD>For vector arguments, return the (real) variance of the values.
For matrix arguments, return a row vector contaning the variance for
each column.
<P>

The argument <VAR>opt</VAR> determines the type of normalization to use. Valid 
values are
</P>
<P>

</P>
<DL COMPACT>
<DT>0:
<DD>  normalizes with N-1, provides the square root of best unbiased estimator
  of the variance [default]
<DT>1:
<DD>  normalizes with N, this provides the square root of the second moment
  around the mean
</DL>
<P>

The third argument <VAR>dim</VAR> determines the dimension along which the 
variance is calculated.
</P>
</DL>
<P>

<A NAME="doc-table"></A>
<A NAME="IDX787"></A>
</P>
<DL>
<DT><U>Function File:</U> [<VAR>t</VAR>, <VAR>l_x</VAR>] = <B>table</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX788"></A>
<DT><U>Function File:</U> [<VAR>t</VAR>, <VAR>l_x</VAR>, <VAR>l_y</VAR>] = <B>table</B> <I>(<VAR>x</VAR>, <VAR>y</VAR>)</I>
<DD>Create a contingency table <VAR>t</VAR> from data vectors.  The <VAR>l</VAR>
vectors are the corresponding levels.
<P>

Currently, only 1- and 2-dimensional tables are supported.
</P>
</DL>
<P>

<A NAME="doc-studentize"></A>
<A NAME="IDX789"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>studentize</B> <I>(<VAR>x</VAR>, <VAR>dim</VAR>)</I>
<DD>If <VAR>x</VAR> is a vector, subtract its mean and divide by its standard
deviation.
<P>

If <VAR>x</VAR> is a matrix, do the above along the first non-singleton
dimension. If the optional argument <VAR>dim</VAR> is given then operate
along this dimension.
</P>
</DL>
<P>

<A NAME="doc-statistics"></A>
<A NAME="IDX790"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>statistics</B> <I>(<VAR>x</VAR>)</I>
<DD>If <VAR>x</VAR> is a matrix, return a matrix with the minimum, first
quartile, median, third quartile, maximum, mean, standard deviation,
skewness and kurtosis of the columns of <VAR>x</VAR> as its rows.
<P>

If <VAR>x</VAR> is a vector, treat it as a column vector.
</P>
</DL>
<P>

<A NAME="doc-spearman"></A>
<A NAME="IDX791"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>spearman</B> <I>(<VAR>x</VAR>, <VAR>y</VAR>)</I>
<DD>Compute Spearman's rank correlation coefficient <VAR>rho</VAR> for each of
the variables specified by the input arguments.
<P>

For matrices, each row is an observation and each column a variable;
vectors are always observations and may be row or column vectors.
</P>
<P>

<CODE>spearman (<VAR>x</VAR>)</CODE> is equivalent to <CODE>spearman (<VAR>x</VAR>,
<VAR>x</VAR>)</CODE>.
</P>
<P>

For two data vectors <VAR>x</VAR> and <VAR>y</VAR>, Spearman's <VAR>rho</VAR> is the
correlation of the ranks of <VAR>x</VAR> and <VAR>y</VAR>.
</P>
<P>

If <VAR>x</VAR> and <VAR>y</VAR> are drawn from independent distributions,
<VAR>rho</VAR> has zero mean and variance <CODE>1 / (n - 1)</CODE>, and is
asymptotically normally distributed.
</P>
</DL>
<P>

<A NAME="doc-run_count"></A>
<A NAME="IDX792"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>run_count</B> <I>(<VAR>x</VAR>, <VAR>n</VAR>)</I>
<DD>Count the upward runs along the first non-singleton dimension of
<VAR>x</VAR> of length 1, 2, ..., <VAR>n</VAR>-1 and greater than or equal 
to <VAR>n</VAR>. If the optional argument <VAR>dim</VAR> is given operate
along this dimension
</DL>
<P>

<A NAME="doc-ranks"></A>
<A NAME="IDX793"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>ranks</B> <I>(<VAR>x</VAR>, <VAR>dim</VAR>)</I>
<DD>If <VAR>x</VAR> is a vector, return the (column) vector of ranks of
<VAR>x</VAR> adjusted for ties.
<P>

If <VAR>x</VAR> is a matrix, do the above for along the first 
non-singleton dimension. If the optional argument <VAR>dim</VAR> is
given, operate along this dimension.
</P>
</DL>
<P>

<A NAME="doc-range"></A>
<A NAME="IDX794"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>range</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX795"></A>
<DT><U>Function File:</U>  <B>range</B> <I>(<VAR>x</VAR>, <VAR>dim</VAR>)</I>
<DD>If <VAR>x</VAR> is a vector, return the range, i.e., the difference
between the maximum and the minimum, of the input data.
<P>

If <VAR>x</VAR> is a matrix, do the above for each column of <VAR>x</VAR>.
</P>
<P>

If the optional argument <VAR>dim</VAR> is supplied, work along dimension
<VAR>dim</VAR>.
</P>
</DL>
<P>

<A NAME="doc-qqplot"></A>
<A NAME="IDX796"></A>
</P>
<DL>
<DT><U>Function File:</U> [<VAR>q</VAR>, <VAR>s</VAR>] = <B>qqplot</B> <I>(<VAR>x</VAR>, <VAR>dist</VAR>, <VAR>params</VAR>)</I>
<DD>Perform a QQ-plot (quantile plot).
<P>

If F is the CDF of the distribution <VAR>dist</VAR> with parameters
<VAR>params</VAR> and G its inverse, and <VAR>x</VAR> a sample vector of length
<VAR>n</VAR>, the QQ-plot graphs ordinate <VAR>s</VAR>(<VAR>i</VAR>) = <VAR>i</VAR>-th
largest element of x versus abscissa <VAR>q</VAR>(<VAR>i</VAR>f) = G((<VAR>i</VAR> -
0.5)/<VAR>n</VAR>).
</P>
<P>

If the sample comes from F except for a transformation of location
and scale, the pairs will approximately follow a straight line.
</P>
<P>

The default for <VAR>dist</VAR> is the standard normal distribution.  The
optional argument <VAR>params</VAR> contains a list of parameters of
<VAR>dist</VAR>.  For example, for a quantile plot of the uniform
distribution on [2,4] and <VAR>x</VAR>, use
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>qqplot (x, &quot;uniform&quot;, 2, 4)
</pre></td></tr></table><P>

If no output arguments are given, the data are plotted directly.
</P>
</DL>
<P>

<A NAME="doc-probit"></A>
<A NAME="IDX797"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>probit</B> <I>(<VAR>p</VAR>)</I>
<DD>For each component of <VAR>p</VAR>, return the probit (the quantile of the
standard normal distribution) of <VAR>p</VAR>.
</DL>
<P>

<A NAME="doc-ppplot"></A>
<A NAME="IDX798"></A>
</P>
<DL>
<DT><U>Function File:</U> [<VAR>p</VAR>, <VAR>y</VAR>] = <B>ppplot</B> <I>(<VAR>x</VAR>, <VAR>dist</VAR>, <VAR>params</VAR>)</I>
<DD>Perform a PP-plot (probability plot).
<P>

If F is the CDF of the distribution <VAR>dist</VAR> with parameters
<VAR>params</VAR> and <VAR>x</VAR> a sample vector of length <VAR>n</VAR>, the
PP-plot graphs ordinate <VAR>y</VAR>(<VAR>i</VAR>) = F (<VAR>i</VAR>-th largest
element of <VAR>x</VAR>) versus abscissa <VAR>p</VAR>(<VAR>i</VAR>) = (<VAR>i</VAR> -
0.5)/<VAR>n</VAR>.  If the sample comes from F, the pairs will
approximately follow a straight line.
</P>
<P>

The default for <VAR>dist</VAR> is the standard normal distribution.  The
optional argument <VAR>params</VAR> contains a list of parameters of
<VAR>dist</VAR>.  For example, for a probability plot of the uniform
distribution on [2,4] and <VAR>x</VAR>, use
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>ppplot (x, &quot;uniform&quot;, 2, 4)
</pre></td></tr></table><P>

If no output arguments are given, the data are plotted directly.
</P>
</DL>
<P>

<A NAME="doc-moment"></A>
<A NAME="IDX799"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>moment</B> <I>(<VAR>x</VAR>, <VAR>p</VAR>, <VAR>opt</VAR>, <VAR>dim</VAR>)</I>
<DD>If <VAR>x</VAR> is a vector, compute the <VAR>p</VAR>-th moment of <VAR>x</VAR>.
<P>

If <VAR>x</VAR> is a matrix, return the row vector containing the
<VAR>p</VAR>-th moment of each column.
</P>
<P>

With the optional string opt, the kind of moment to be computed can
be specified.  If opt contains <CODE>&quot;c&quot;</CODE> or <CODE>&quot;a&quot;</CODE>, central
and/or absolute moments are returned.  For example,
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>moment (x, 3, &quot;ac&quot;)
</pre></td></tr></table><P>

computes the third central absolute moment of <VAR>x</VAR>.
</P>
<P>

If the optional argument <VAR>dim</VAR> is supplied, work along dimension
<VAR>dim</VAR>.
</P>
</DL>
<P>

<A NAME="doc-meansq"></A>
<A NAME="IDX800"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>meansq</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX801"></A>
<DT><U>Function File:</U>  <B>meansq</B> <I>(<VAR>x</VAR>, <VAR>dim</VAR>)</I>
<DD>For vector arguments, return the mean square of the values.
For matrix arguments, return a row vector contaning the mean square
of each column. With the optional <VAR>dim</VAR> argument, returns the
mean squared of the values along this dimension
</DL>
<P>

<A NAME="doc-logit"></A>
<A NAME="IDX802"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>logit</B> <I>(<VAR>p</VAR>)</I>
<DD>For each component of <VAR>p</VAR>, return the logit <CODE>log (<VAR>p</VAR> /
(1-<VAR>p</VAR>))</CODE> of <VAR>p</VAR>.
</DL>
<P>

<A NAME="doc-kendall"></A>
<A NAME="IDX803"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>kendall</B> <I>(<VAR>x</VAR>, <VAR>y</VAR>)</I>
<DD>Compute Kendall's <VAR>tau</VAR> for each of the variables specified by
the input arguments.
<P>

For matrices, each row is an observation and each column a variable;
vectors are always observations and may be row or column vectors.
</P>
<P>

<CODE>kendall (<VAR>x</VAR>)</CODE> is equivalent to <CODE>kendall (<VAR>x</VAR>,
<VAR>x</VAR>)</CODE>.
</P>
<P>

For two data vectors <VAR>x</VAR>, <VAR>y</VAR> of common length <VAR>n</VAR>,
Kendall's <VAR>tau</VAR> is the correlation of the signs of all rank
differences of <VAR>x</VAR> and <VAR>y</VAR>;  i.e., if both <VAR>x</VAR> and
<VAR>y</VAR> have distinct entries, then
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>         1    
tau = -------   SUM sign (q(i) - q(j)) * sign (r(i) - r(j))
      n (n-1)   i,j
</pre></td></tr></table><P>

in which the
<VAR>q</VAR>(<VAR>i</VAR>) and <VAR>r</VAR>(<VAR>i</VAR>)
 are the ranks of
<VAR>x</VAR> and <VAR>y</VAR>, respectively.
</P>
<P>

If <VAR>x</VAR> and <VAR>y</VAR> are drawn from independent distributions,
Kendall's <VAR>tau</VAR> is asymptotically normal with mean 0 and variance
<CODE>(2 * (2<VAR>n</VAR>+5)) / (9 * <VAR>n</VAR> * (<VAR>n</VAR>-1))</CODE>.
</P>
</DL>
<P>

<A NAME="doc-iqr"></A>
<A NAME="IDX804"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>iqr</B> <I>(<VAR>x</VAR>, <VAR>dim</VAR>)</I>
<DD>If <VAR>x</VAR> is a vector, return the interquartile range, i.e., the
difference between the upper and lower quartile, of the input data.
<P>

If <VAR>x</VAR> is a matrix, do the above for first non singleton
dimension of <VAR>x</VAR>.. If the option <VAR>dim</VAR> argument is given,
then operate along this dimension.
</P>
</DL>
<P>

<A NAME="doc-cut"></A>
<A NAME="IDX805"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>cut</B> <I>(<VAR>x</VAR>, <VAR>breaks</VAR>)</I>
<DD>Create categorical data out of numerical or continuous data by
cutting into intervals.
<P>

If <VAR>breaks</VAR> is a scalar, the data is cut into that many
equal-width intervals.  If <VAR>breaks</VAR> is a vector of break points,
the category has <CODE>length (<VAR>breaks</VAR>) - 1</CODE> groups.
</P>
<P>

The returned value is a vector of the same size as <VAR>x</VAR> telling
which group each point in <VAR>x</VAR> belongs to.  Groups are labelled
from 1 to the number of groups; points outside the range of
<VAR>breaks</VAR> are labelled by <CODE>NaN</CODE>.
</P>
</DL>
<P>

<A NAME="doc-cor"></A>
<A NAME="IDX806"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>cor</B> <I>(<VAR>x</VAR>, <VAR>y</VAR>)</I>
<DD>The (<VAR>i</VAR>, <VAR>j</VAR>)-th entry of <CODE>cor (<VAR>x</VAR>, <VAR>y</VAR>)</CODE> is
the correlation between the <VAR>i</VAR>-th variable in <VAR>x</VAR> and the
<VAR>j</VAR>-th variable in <VAR>y</VAR>.
<P>

For matrices, each row is an observation and each column a variable;
vectors are always observations and may be row or column vectors.
</P>
<P>

<CODE>cor (<VAR>x</VAR>)</CODE> is equivalent to <CODE>cor (<VAR>x</VAR>, <VAR>x</VAR>)</CODE>.
</P>
</DL>
<P>

<A NAME="doc-cloglog"></A>
<A NAME="IDX807"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>cloglog</B> <I>(<VAR>x</VAR>)</I>
<DD>Return the complementary log-log function of <VAR>x</VAR>, defined as
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>- log (- log (<VAR>x</VAR>))
</pre></td></tr></table></DL>
<P>

<A NAME="doc-center"></A>
<A NAME="IDX808"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>center</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX809"></A>
<DT><U>Function File:</U>  <B>center</B> <I>(<VAR>x</VAR>, <VAR>dim</VAR>)</I>
<DD>If <VAR>x</VAR> is a vector, subtract its mean.
If <VAR>x</VAR> is a matrix, do the above for each column.
If the optional argument <VAR>dim</VAR> is given, perform the above
operation along this dimension
</DL>
<P>

<A NAME="Tests"></A>
<HR SIZE="6">
<A NAME="SEC178"></A>
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</TR></TABLE>
<H2> 25.2 Tests </H2>
<!--docid::SEC178::-->
<P>

<A NAME="doc-anova"></A>
<A NAME="IDX810"></A>
</P>
<DL>
<DT><U>Function File:</U> [<VAR>pval</VAR>, <VAR>f</VAR>, <VAR>df_b</VAR>, <VAR>df_w</VAR>] = <B>anova</B> <I>(<VAR>y</VAR>, <VAR>g</VAR>)</I>
<DD>Perform a one-way analysis of variance (ANOVA).  The goal is to test
whether the population means of data taken from <VAR>k</VAR> different
groups are all equal.
<P>

Data may be given in a single vector <VAR>y</VAR> with groups specified by
a corresponding vector of group labels <VAR>g</VAR> (e.g., numbers from 1
to <VAR>k</VAR>). This is the general form which does not impose any
restriction on the number of data in each group or the group labels.
</P>
<P>

If <VAR>y</VAR> is a matrix and <VAR>g</VAR> is omitted, each column of <VAR>y</VAR>
is treated as a group.  This form is only appropriate for balanced
ANOVA in which the numbers of samples from each group are all equal.
</P>
<P>

Under the null of constant means, the statistic <VAR>f</VAR> follows an F
distribution with <VAR>df_b</VAR> and <VAR>df_w</VAR> degrees of freedom.
</P>
<P>

The p-value (1 minus the CDF of this distribution at <VAR>f</VAR>) is
returned in <VAR>pval</VAR>.
</P>
<P>

If no output argument is given, the standard one-way ANOVA table is
printed.
</P>
</DL>
<P>

<A NAME="doc-bartlett_test"></A>
<A NAME="IDX811"></A>
</P>
<DL>
<DT><U>Function File:</U> [<VAR>pval</VAR>, <VAR>chisq</VAR>, <VAR>df</VAR>] = <B>bartlett_test</B> <I>(<VAR>x1</VAR>, <small>...</small>)</I>
<DD>Perform a Bartlett test for the homogeneity of variances in the data
vectors <VAR>x1</VAR>, <VAR>x2</VAR>, <small>...</small>, <VAR>xk</VAR>, where <VAR>k</VAR> &gt; 1.
<P>

Under the null of equal variances, the test statistic <VAR>chisq</VAR>
approximately ollows a chi-square distribution with <VAR>df</VAR> degrees of
freedom.
</P>
<P>

The p-value (1 minus the CDF of this distribution at <VAR>chisq</VAR>) is
returned in <VAR>pval</VAR>.
</P>
<P>

If no output argument is given, the p-value is displayed.
</P>
</DL>
<P>

<A NAME="doc-chisquare_test_homogeneity"></A>
<A NAME="IDX812"></A>
</P>
<DL>
<DT><U>Function File:</U> [<VAR>pval</VAR>, <VAR>chisq</VAR>, <VAR>df</VAR>] = <B>chisquare_test_homogeneity</B> <I>(<VAR>x</VAR>, <VAR>y</VAR>, <VAR>c</VAR>)</I>
<DD>Given two samples <VAR>x</VAR> and <VAR>y</VAR>, perform a chisquare test for
homogeneity of the null hypothesis that <VAR>x</VAR> and <VAR>y</VAR> come from
the same distribution, based on the partition induced by the
(strictly increasing) entries of <VAR>c</VAR>.
<P>

For large samples, the test statistic <VAR>chisq</VAR> approximately follows a
chisquare distribution with <VAR>df</VAR> = <CODE>length (<VAR>c</VAR>)</CODE>
degrees of freedom.
</P>
<P>

The p-value (1 minus the CDF of this distribution at <VAR>chisq</VAR>) is
returned in <VAR>pval</VAR>.
</P>
<P>

If no output argument is given, the p-value is displayed.
</P>
</DL>
<P>

<A NAME="doc-chisquare_test_independence"></A>
<A NAME="IDX813"></A>
</P>
<DL>
<DT><U>Function File:</U> [<VAR>pval</VAR>, <VAR>chisq</VAR>, <VAR>df</VAR>] = <B>chisquare_test_independence</B> <I>(<VAR>x</VAR>)</I>
<DD>Perform a chi-square test for indepence based on the contingency
table <VAR>x</VAR>.  Under the null hypothesis of independence,
<VAR>chisq</VAR> approximately has a chi-square distribution with
<VAR>df</VAR> degrees of freedom.
<P>

The p-value (1 minus the CDF of this distribution at chisq) of the
test is returned in <VAR>pval</VAR>.
</P>
<P>

If no output argument is given, the p-value is displayed.
</P>
</DL>
<P>

<A NAME="doc-cor_test"></A>
<A NAME="IDX814"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>cor_test</B> <I>(<VAR>x</VAR>, <VAR>y</VAR>, <VAR>alt</VAR>, <VAR>method</VAR>)</I>
<DD>Test whether two samples <VAR>x</VAR> and <VAR>y</VAR> come from uncorrelated
populations.
<P>

The optional argument string <VAR>alt</VAR> describes the alternative
hypothesis, and can be <CODE>&quot;!=&quot;</CODE> or <CODE>&quot;&lt;&gt;&quot;</CODE> (non-zero),
<CODE>&quot;&gt;&quot;</CODE> (greater than 0), or <CODE>&quot;&lt;&quot;</CODE> (less than 0).  The
default is the two-sided case.
</P>
<P>

The optional argument string <VAR>method</VAR> specifies on which
correlation coefficient the test should be based.  If <VAR>method</VAR> is
<CODE>&quot;pearson&quot;</CODE> (default), the (usual) Pearson's product moment
correlation coefficient is used.  In this case, the data should come
from a bivariate normal distribution.  Otherwise, the other two
methods offer nonparametric alternatives. If <VAR>method</VAR> is
<CODE>&quot;kendall&quot;</CODE>, then Kendall's rank correlation tau is used.  If
<VAR>method</VAR> is <CODE>&quot;spearman&quot;</CODE>, then Spearman's rank correlation
rho is used.  Only the first character is necessary.
</P>
<P>

The output is a structure with the following elements:
</P>
<P>

</P>
<DL COMPACT>
<DT><VAR>pval</VAR>
<DD>The p-value of the test.
<DT><VAR>stat</VAR>
<DD>The value of the test statistic.
<DT><VAR>dist</VAR>
<DD>The distribution of the test statistic.
<DT><VAR>params</VAR>
<DD>The parameters of the null distribution of the test statistic.
<DT><VAR>alternative</VAR>
<DD>The alternative hypothesis.
<DT><VAR>method</VAR>
<DD>The method used for testing.
</DL>
<P>

If no output argument is given, the p-value is displayed.
</P>
</DL>
<P>

<A NAME="doc-f_test_regression"></A>
<A NAME="IDX815"></A>
</P>
<DL>
<DT><U>Function File:</U> [<VAR>pval</VAR>, <VAR>f</VAR>, <VAR>df_num</VAR>, <VAR>df_den</VAR>] = <B>f_test_regression</B> <I>(<VAR>y</VAR>, <VAR>x</VAR>, <VAR>rr</VAR>, <VAR>r</VAR>)</I>
<DD>Perform an F test for the null hypothesis rr * b = r in a classical
normal regression model y = X * b + e.
<P>

Under the null, the test statistic <VAR>f</VAR> follows an F distribution
with <VAR>df_num</VAR> and <VAR>df_den</VAR> degrees of freedom.
</P>
<P>

The p-value (1 minus the CDF of this distribution at <VAR>f</VAR>) is
returned in <VAR>pval</VAR>.
</P>
<P>

If not given explicitly, <VAR>r</VAR> = 0.
</P>
<P>

If no output argument is given, the p-value is displayed.
</P>
</DL>
<P>

<A NAME="doc-hotelling_test"></A>
<A NAME="IDX816"></A>
</P>
<DL>
<DT><U>Function File:</U> [<VAR>pval</VAR>, <VAR>tsq</VAR>] = <B>hotelling_test</B> <I>(<VAR>x</VAR>, <VAR>m</VAR>)</I>
<DD>For a sample <VAR>x</VAR> from a multivariate normal distribution with unknown
mean and covariance matrix, test the null hypothesis that <CODE>mean
(<VAR>x</VAR>) == <VAR>m</VAR></CODE>.
<P>

Hotelling's T^2 is returned in <VAR>tsq</VAR>.  Under the null,
<EM></EM> has an F distribution with <EM></EM> and
<EM></EM> degrees of freedom, where <EM></EM> and <EM></EM> are the
numbers of samples and variables, respectively.
</P>
<P>

The p-value of the test is returned in <VAR>pval</VAR>.
</P>
<P>

If no output argument is given, the p-value of the test is displayed.
</P>
</DL>
<P>

<A NAME="doc-hotelling_test_2"></A>
<A NAME="IDX817"></A>
</P>
<DL>
<DT><U>Function File:</U> [<VAR>pval</VAR>, <VAR>tsq</VAR>] = <B>hotelling_test_2</B> <I>(<VAR>x</VAR>, <VAR>y</VAR>)</I>
<DD>For two samples <VAR>x</VAR> from multivariate normal distributions with
the same number of variables (columns), unknown means and unknown
equal covariance matrices, test the null hypothesis <CODE>mean
(<VAR>x</VAR>) == mean (<VAR>y</VAR>)</CODE>.
<P>

Hotelling's two-sample T^2 is returned in <VAR>tsq</VAR>.  Under the null,
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>(n_x+n_y-p-1) T^2 / (p(n_x+n_y-2))
</pre></td></tr></table><P>

has an F distribution with <EM></EM> and <EM></EM> degrees of
freedom, where <EM></EM> and <EM></EM> are the sample sizes and
<EM></EM> is the number of variables.
</P>
<P>

The p-value of the test is returned in <VAR>pval</VAR>.
</P>
<P>

If no output argument is given, the p-value of the test is displayed.
</P>
</DL>
<P>

<A NAME="doc-kolmogorov_smirnov_test"></A>
<A NAME="IDX818"></A>
</P>
<DL>
<DT><U>Function File:</U> [<VAR>pval</VAR>, <VAR>ks</VAR>] = <B>kolmogorov_smirnov_test</B> <I>(<VAR>x</VAR>, <VAR>dist</VAR>, <VAR>params</VAR>, <VAR>alt</VAR>)</I>
<DD>Perform a Kolmogorov-Smirnov test of the null hypothesis that the
sample <VAR>x</VAR> comes from the (continuous) distribution dist. I.e.,
if F and G are the CDFs corresponding to the sample and dist,
respectively, then the null is that F == G.
<P>

The optional argument <VAR>params</VAR> contains a list of parameters of
<VAR>dist</VAR>.  For example, to test whether a sample <VAR>x</VAR> comes from
a uniform distribution on [2,4], use
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>kolmogorov_smirnov_test(x, &quot;uniform&quot;, 2, 4)
</pre></td></tr></table><P>

With the optional argument string <VAR>alt</VAR>, the alternative of
interest can be selected.  If <VAR>alt</VAR> is <CODE>&quot;!=&quot;</CODE> or
<CODE>&quot;&lt;&gt;&quot;</CODE>, the null is tested against the two-sided alternative F
!= G.  In this case, the test statistic <VAR>ks</VAR> follows a two-sided
Kolmogorov-Smirnov distribution.  If <VAR>alt</VAR> is <CODE>&quot;&gt;&quot;</CODE>, the
one-sided alternative F &gt; G is considered.  Similarly for <CODE>&quot;&lt;&quot;</CODE>,
the one-sided alternative F &gt; G is considered.  In this case, the
test statistic <VAR>ks</VAR> has a one-sided Kolmogorov-Smirnov
distribution.  The default is the two-sided case.
</P>
<P>

The p-value of the test is returned in <VAR>pval</VAR>.
</P>
<P>

If no output argument is given, the p-value is displayed.
</P>
</DL>
<P>

<A NAME="doc-kolmogorov_smirnov_test_2"></A>
<A NAME="IDX819"></A>
</P>
<DL>
<DT><U>Function File:</U> [<VAR>pval</VAR>, <VAR>ks</VAR>, <VAR>d</VAR>] = <B>kolmogorov_smirnov_test_2</B> <I>(<VAR>x</VAR>, <VAR>y</VAR>, <VAR>alt</VAR>)</I>
<DD>Perform a 2-sample Kolmogorov-Smirnov test of the null hypothesis
that the samples <VAR>x</VAR> and <VAR>y</VAR> come from the same (continuous)
distribution.  I.e., if F and G are the CDFs corresponding to the
<VAR>x</VAR> and <VAR>y</VAR> samples, respectively, then the null is that F ==
G.
<P>

With the optional argument string <VAR>alt</VAR>, the alternative of
interest can be selected.  If <VAR>alt</VAR> is <CODE>&quot;!=&quot;</CODE> or
<CODE>&quot;&lt;&gt;&quot;</CODE>, the null is tested against the two-sided alternative F
!= G.  In this case, the test statistic <VAR>ks</VAR> follows a two-sided
Kolmogorov-Smirnov distribution.  If <VAR>alt</VAR> is <CODE>&quot;&gt;&quot;</CODE>, the
one-sided alternative F &gt; G is considered.  Similarly for <CODE>&quot;&lt;&quot;</CODE>,
the one-sided alternative F &lt; G is considered.  In this case, the
test statistic <VAR>ks</VAR> has a one-sided Kolmogorov-Smirnov
distribution.  The default is the two-sided case.
</P>
<P>

The p-value of the test is returned in <VAR>pval</VAR>.
</P>
<P>

The third returned value, <VAR>d</VAR>, is the test statistic, the maximum
vertical distance between the two cumulative distribution functions.
</P>
<P>

If no output argument is given, the p-value is displayed.
</P>
</DL>
<P>

<A NAME="doc-kruskal_wallis_test"></A>
<A NAME="IDX820"></A>
</P>
<DL>
<DT><U>Function File:</U> [<VAR>pval</VAR>, <VAR>k</VAR>, <VAR>df</VAR>] = <B>kruskal_wallis_test</B> <I>(<VAR>x1</VAR>, <small>...</small>)</I>
<DD>Perform a Kruskal-Wallis one-factor &quot;analysis of variance&quot;.
<P>

Suppose a variable is observed for <VAR>k</VAR> &gt; 1 different groups, and
let <VAR>x1</VAR>, <small>...</small>, <VAR>xk</VAR> be the corresponding data vectors.
</P>
<P>

Under the null hypothesis that the ranks in the pooled sample are not
affected by the group memberships, the test statistic <VAR>k</VAR> is
approximately chi-square with <VAR>df</VAR> = <VAR>k</VAR> - 1 degrees of
freedom.
</P>
<P>

The p-value (1 minus the CDF of this distribution at <VAR>k</VAR>) is
returned in <VAR>pval</VAR>.
</P>
<P>

If no output argument is given, the p-value is displayed.
</P>
</DL>
<P>

<A NAME="doc-manova"></A>
<A NAME="IDX821"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>manova</B> <I>(<VAR>y</VAR>, <VAR>g</VAR>)</I>
<DD>Perform a one-way multivariate analysis of variance (MANOVA). The
goal is to test whether the p-dimensional population means of data
taken from <VAR>k</VAR> different groups are all equal.  All data are
assumed drawn independently from p-dimensional normal distributions
with the same covariance matrix.
<P>

The data matrix is given by <VAR>y</VAR>.  As usual, rows are observations
and columns are variables.  The vector <VAR>g</VAR> specifies the
corresponding group labels (e.g., numbers from 1 to <VAR>k</VAR>).
</P>
<P>

The LR test statistic (Wilks' Lambda) and approximate p-values are
computed and displayed.
</P>
</DL>
<P>

<A NAME="doc-mcnemar_test"></A>
<A NAME="IDX822"></A>
</P>
<DL>
<DT><U>Function File:</U> [<VAR>pval</VAR>, <VAR>chisq</VAR>, <VAR>df</VAR>] = <B>mcnemar_test</B> <I>(<VAR>x</VAR>)</I>
<DD>For a square contingency table <VAR>x</VAR> of data cross-classified on
the row and column variables, McNemar's test can be used for testing
the null hypothesis of symmetry of the classification probabilities.
<P>

Under the null, <VAR>chisq</VAR> is approximately distributed as chisquare
with <VAR>df</VAR> degrees of freedom.
</P>
<P>

The p-value (1 minus the CDF of this distribution at <VAR>chisq</VAR>) is
returned in <VAR>pval</VAR>.
</P>
<P>

If no output argument is given, the p-value of the test is displayed.
</P>
</DL>
<P>

<A NAME="doc-prop_test_2"></A>
<A NAME="IDX823"></A>
</P>
<DL>
<DT><U>Function File:</U> [<VAR>pval</VAR>, <VAR>z</VAR>] = <B>prop_test_2</B> <I>(<VAR>x1</VAR>, <VAR>n1</VAR>, <VAR>x2</VAR>, <VAR>n2</VAR>, <VAR>alt</VAR>)</I>
<DD>If <VAR>x1</VAR> and <VAR>n1</VAR> are the counts of successes and trials in
one sample, and <VAR>x2</VAR> and <VAR>n2</VAR> those in a second one, test the
null hypothesis that the success probabilities <VAR>p1</VAR> and <VAR>p2</VAR>
are the same.  Under the null, the test statistic <VAR>z</VAR>
approximately follows a standard normal distribution.
<P>

With the optional argument string <VAR>alt</VAR>, the alternative of
interest can be selected.  If <VAR>alt</VAR> is <CODE>&quot;!=&quot;</CODE> or
<CODE>&quot;&lt;&gt;&quot;</CODE>, the null is tested against the two-sided alternative
<VAR>p1</VAR> != <VAR>p2</VAR>.  If <VAR>alt</VAR> is <CODE>&quot;&gt;&quot;</CODE>, the one-sided
alternative <VAR>p1</VAR> &gt; <VAR>p2</VAR> is used.  Similarly for <CODE>&quot;&lt;&quot;</CODE>,
the one-sided alternative <VAR>p1</VAR> &lt; <VAR>p2</VAR> is used.
The default is the two-sided case.
</P>
<P>

The p-value of the test is returned in <VAR>pval</VAR>.
</P>
<P>

If no output argument is given, the p-value of the test is displayed.
</P>
</DL>
<P>

<A NAME="doc-run_test"></A>
<A NAME="IDX824"></A>
</P>
<DL>
<DT><U>Function File:</U> [<VAR>pval</VAR>, <VAR>chisq</VAR>] = <B>run_test</B> <I>(<VAR>x</VAR>)</I>
<DD>Perform a chi-square test with 6 degrees of freedom based on the
upward runs in the columns of <VAR>x</VAR>.  Can be used to test whether
<VAR>x</VAR> contains independent data.
<P>

The p-value of the test is returned in <VAR>pval</VAR>.
</P>
<P>

If no output argument is given, the p-value is displayed.
</P>
</DL>
<P>

<A NAME="doc-sign_test"></A>
<A NAME="IDX825"></A>
</P>
<DL>
<DT><U>Function File:</U> [<VAR>pval</VAR>, <VAR>b</VAR>, <VAR>n</VAR>] = <B>sign_test</B> <I>(<VAR>x</VAR>, <VAR>y</VAR>, <VAR>alt</VAR>)</I>
<DD>For two matched-pair samples <VAR>x</VAR> and <VAR>y</VAR>, perform a sign test
of the null hypothesis PROB (<VAR>x</VAR> &gt; <VAR>y</VAR>) == PROB (<VAR>x</VAR> &lt;
<VAR>y</VAR>) == 1/2.  Under the null, the test statistic <VAR>b</VAR> roughly
follows a binomial distribution with parameters <CODE><VAR>n</VAR> = sum
(<VAR>x</VAR> != <VAR>y</VAR>)</CODE> and <VAR>p</VAR> = 1/2.
<P>

With the optional argument <CODE>alt</CODE>, the alternative of interest
can be selected.  If <VAR>alt</VAR> is <CODE>&quot;!=&quot;</CODE> or <CODE>&quot;&lt;&gt;&quot;</CODE>, the
null hypothesis is tested against the two-sided alternative PROB
(<VAR>x</VAR> &lt; <VAR>y</VAR>) != 1/2.  If <VAR>alt</VAR> is <CODE>&quot;&gt;&quot;</CODE>, the
one-sided alternative PROB (<VAR>x</VAR> &gt; <VAR>y</VAR>) &gt; 1/2 (&quot;x is
stochastically greater than y&quot;) is considered.  Similarly for
<CODE>&quot;&lt;&quot;</CODE>, the one-sided alternative PROB (<VAR>x</VAR> &gt; <VAR>y</VAR>) &lt; 1/2
(&quot;x is stochastically less than y&quot;) is considered.  The default is
the two-sided case.
</P>
<P>

The p-value of the test is returned in <VAR>pval</VAR>.
</P>
<P>

If no output argument is given, the p-value of the test is displayed.
</P>
</DL>
<P>

<A NAME="doc-t_test"></A>
<A NAME="IDX826"></A>
</P>
<DL>
<DT><U>Function File:</U> [<VAR>pval</VAR>, <VAR>t</VAR>, <VAR>df</VAR>] = <B>t_test</B> <I>(<VAR>x</VAR>, <VAR>m</VAR>, <VAR>alt</VAR>)</I>
<DD>For a sample <VAR>x</VAR> from a normal distribution with unknown mean and
variance, perform a t-test of the null hypothesis <CODE>mean
(<VAR>x</VAR>) == <VAR>m</VAR></CODE>.  Under the null, the test statistic <VAR>t</VAR>
follows a Student distribution with <CODE><VAR>df</VAR> = length (<VAR>x</VAR>)
- 1</CODE> degrees of freedom.
<P>

With the optional argument string <VAR>alt</VAR>, the alternative of
interest can be selected.  If <VAR>alt</VAR> is <CODE>&quot;!=&quot;</CODE> or
<CODE>&quot;&lt;&gt;&quot;</CODE>, the null is tested against the two-sided alternative
<CODE>mean (<VAR>x</VAR>) != <VAR>m</VAR></CODE>.  If <VAR>alt</VAR> is <CODE>&quot;&gt;&quot;</CODE>, the
one-sided alternative <CODE>mean (<VAR>x</VAR>) &gt; <VAR>m</VAR></CODE> is considered.
Similarly for <VAR>&quot;&lt;&quot;</VAR>, the one-sided alternative <CODE>mean
(<VAR>x</VAR>) &lt; <VAR>m</VAR></CODE> is considered,  The default is the two-sided
case.
</P>
<P>

The p-value of the test is returned in <VAR>pval</VAR>.
</P>
<P>

If no output argument is given, the p-value of the test is displayed.
</P>
</DL>
<P>

<A NAME="doc-t_test_2"></A>
<A NAME="IDX827"></A>
</P>
<DL>
<DT><U>Function File:</U> [<VAR>pval</VAR>, <VAR>t</VAR>, <VAR>df</VAR>] = <B>t_test_2</B> <I>(<VAR>x</VAR>, <VAR>y</VAR>, <VAR>alt</VAR>)</I>
<DD>For two samples x and y from normal distributions with unknown means
and unknown equal variances, perform a two-sample t-test of the null
hypothesis of equal means.  Under the null, the test statistic
<VAR>t</VAR> follows a Student distribution with <VAR>df</VAR> degrees of
freedom.
<P>

With the optional argument string <VAR>alt</VAR>, the alternative of
interest can be selected.  If <VAR>alt</VAR> is <CODE>&quot;!=&quot;</CODE> or
<CODE>&quot;&lt;&gt;&quot;</CODE>, the null is tested against the two-sided alternative
<CODE>mean (<VAR>x</VAR>) != mean (<VAR>y</VAR>)</CODE>.  If <VAR>alt</VAR> is <CODE>&quot;&gt;&quot;</CODE>,
the one-sided alternative <CODE>mean (<VAR>x</VAR>) &gt; mean (<VAR>y</VAR>)</CODE> is
used.  Similarly for <CODE>&quot;&lt;&quot;</CODE>, the one-sided alternative <CODE>mean
(<VAR>x</VAR>) &lt; mean (<VAR>y</VAR>)</CODE> is used.  The default is the two-sided
case.
</P>
<P>

The p-value of the test is returned in <VAR>pval</VAR>.
</P>
<P>

If no output argument is given, the p-value of the test is displayed.
</P>
</DL>
<P>

<A NAME="doc-t_test_regression"></A>
<A NAME="IDX828"></A>
</P>
<DL>
<DT><U>Function File:</U> [<VAR>pval</VAR>, <VAR>t</VAR>, <VAR>df</VAR>] = <B>t_test_regression</B> <I>(<VAR>y</VAR>, <VAR>x</VAR>, <VAR>rr</VAR>, <VAR>r</VAR>, <VAR>alt</VAR>)</I>
<DD>Perform an t test for the null hypothesis <CODE><VAR>rr</VAR> * <VAR>b</VAR> =
<VAR>r</VAR></CODE> in a classical normal regression model <CODE><VAR>y</VAR> =
<VAR>x</VAR> * <VAR>b</VAR> + <VAR>e</VAR></CODE>.  Under the null, the test statistic <VAR>t</VAR>
follows a <VAR>t</VAR> distribution with <VAR>df</VAR> degrees of freedom.
<P>

If <VAR>r</VAR> is omitted, a value of 0 is assumed.
</P>
<P>

With the optional argument string <VAR>alt</VAR>, the alternative of
interest can be selected.  If <VAR>alt</VAR> is <CODE>&quot;!=&quot;</CODE> or
<CODE>&quot;&lt;&gt;&quot;</CODE>, the null is tested against the two-sided alternative
<CODE><VAR>rr</VAR> * <VAR>b</VAR> != <VAR>r</VAR></CODE>.  If <VAR>alt</VAR> is <CODE>&quot;&gt;&quot;</CODE>, the
one-sided alternative <CODE><VAR>rr</VAR> * <VAR>b</VAR> &gt; <VAR>r</VAR></CODE> is used.
Similarly for <VAR>&quot;&lt;&quot;</VAR>, the one-sided alternative <CODE><VAR>rr</VAR> *
<VAR>b</VAR> &lt; <VAR>r</VAR></CODE> is used.  The default is the two-sided case. 
</P>
<P>

The p-value of the test is returned in <VAR>pval</VAR>.
</P>
<P>

If no output argument is given, the p-value of the test is displayed.
</P>
</DL>
<P>

<A NAME="doc-u_test"></A>
<A NAME="IDX829"></A>
</P>
<DL>
<DT><U>Function File:</U> [<VAR>pval</VAR>, <VAR>z</VAR>] = <B>u_test</B> <I>(<VAR>x</VAR>, <VAR>y</VAR>, <VAR>alt</VAR>)</I>
<DD>For two samples <VAR>x</VAR> and <VAR>y</VAR>, perform a Mann-Whitney U-test of
the null hypothesis PROB (<VAR>x</VAR> &gt; <VAR>y</VAR>) == 1/2 == PROB (<VAR>x</VAR>
&lt; <VAR>y</VAR>).  Under the null, the test statistic <VAR>z</VAR> approximately
follows a standard normal distribution.  Note that this test is
equivalent to the Wilcoxon rank-sum test.
<P>

With the optional argument string <VAR>alt</VAR>, the alternative of
interest can be selected.  If <VAR>alt</VAR> is <CODE>&quot;!=&quot;</CODE> or
<CODE>&quot;&lt;&gt;&quot;</CODE>, the null is tested against the two-sided alternative
PROB (<VAR>x</VAR> &gt; <VAR>y</VAR>) != 1/2.  If <VAR>alt</VAR> is <CODE>&quot;&gt;&quot;</CODE>, the
one-sided alternative PROB (<VAR>x</VAR> &gt; <VAR>y</VAR>) &gt; 1/2 is considered.
Similarly for <CODE>&quot;&lt;&quot;</CODE>, the one-sided alternative PROB (<VAR>x</VAR> &gt;
<VAR>y</VAR>) &lt; 1/2 is considered,  The default is the two-sided case.
</P>
<P>

The p-value of the test is returned in <VAR>pval</VAR>.
</P>
<P>

If no output argument is given, the p-value of the test is displayed.
</P>
</DL>
<P>

<A NAME="doc-var_test"></A>
<A NAME="IDX830"></A>
</P>
<DL>
<DT><U>Function File:</U> [<VAR>pval</VAR>, <VAR>f</VAR>, <VAR>df_num</VAR>, <VAR>df_den</VAR>] = <B>var_test</B> <I>(<VAR>x</VAR>, <VAR>y</VAR>, <VAR>alt</VAR>)</I>
<DD>For two samples <VAR>x</VAR> and <VAR>y</VAR> from normal distributions with
unknown means and unknown variances, perform an F-test of the null
hypothesis of equal variances.  Under the null, the test statistic f
follows an F-distribution with df_num and df_den degrees of freedom.
<P>

With the optional argument string <VAR>alt</VAR>, the alternative of
interest can be selected.  If <VAR>alt</VAR> is <CODE>&quot;!=&quot;</CODE> or
<CODE>&quot;&lt;&gt;&quot;</CODE>, the null is tested against the two-sided alternative
<CODE>var (<VAR>x</VAR>) != var (<VAR>y</VAR>)</CODE>.  If <VAR>alt</VAR> is <CODE>&quot;&gt;&quot;</CODE>,
the one-sided alternative <CODE>var (<VAR>x</VAR>) &gt; var (<VAR>y</VAR>)</CODE> is
used.  Similarly for &quot;&lt;&quot;, the one-sided alternative <CODE>var
(<VAR>x</VAR>) &gt; var (<VAR>y</VAR>)</CODE> is used.  The default is the two-sided
case.
</P>
<P>

The p-value of the test is returned in <VAR>pval</VAR>.
</P>
<P>

If no output argument is given, the p-value of the test is displayed.
</P>
</DL>
<P>

<A NAME="doc-welch_test"></A>
<A NAME="IDX831"></A>
</P>
<DL>
<DT><U>Function File:</U> [<VAR>pval</VAR>, <VAR>t</VAR>, <VAR>df</VAR>] = <B>welch_test</B> <I>(<VAR>x</VAR>, <VAR>y</VAR>, <VAR>alt</VAR>)</I>
<DD>For two samples <VAR>x</VAR> and <VAR>y</VAR> from normal distributions with
unknown means and unknown and not necessarily equal variances,
perform a Welch test of the null hypothesis of equal means.
Under the null, the test statistic t approximately follows a Student
distribution with df degrees of freedom.
<P>

With the optional argument string <VAR>alt</VAR>, the alternative of
interest can be selected.  If <VAR>alt</VAR> is <CODE>&quot;!=&quot;</CODE> or
<CODE>&quot;&lt;&gt;&quot;</CODE>, the null is tested against the two-sided alternative
<CODE>mean (<VAR>x</VAR>) != <VAR>m</VAR></CODE>.  If <VAR>alt</VAR> is <CODE>&quot;&gt;&quot;</CODE>, the
one-sided alternative mean(x) &gt; <VAR>m</VAR> is considered.  Similarly for
<CODE>&quot;&lt;&quot;</CODE>, the one-sided alternative mean(x) &lt; <VAR>m</VAR> is
considered.  The default is the two-sided case.
</P>
<P>

The p-value of the test is returned in <VAR>pval</VAR>.
</P>
<P>

If no output argument is given, the p-value of the test is displayed.
</P>
</DL>
<P>

<A NAME="doc-wilcoxon_test"></A>
<A NAME="IDX832"></A>
</P>
<DL>
<DT><U>Function File:</U> [<VAR>pval</VAR>, <VAR>z</VAR>] = <B>wilcoxon_test</B> <I>(<VAR>x</VAR>, <VAR>y</VAR>, <VAR>alt</VAR>)</I>
<DD>For two matched-pair sample vectors <VAR>x</VAR> and <VAR>y</VAR>, perform a
Wilcoxon signed-rank test of the null hypothesis PROB (<VAR>x</VAR> &gt;
<VAR>y</VAR>) == 1/2.  Under the null, the test statistic <VAR>z</VAR>
approximately follows a standard normal distribution.
<P>

With the optional argument string <VAR>alt</VAR>, the alternative of
interest can be selected.  If <VAR>alt</VAR> is <CODE>&quot;!=&quot;</CODE> or
<CODE>&quot;&lt;&gt;&quot;</CODE>, the null is tested against the two-sided alternative
PROB (<VAR>x</VAR> &gt; <VAR>y</VAR>) != 1/2.  If alt is <CODE>&quot;&gt;&quot;</CODE>, the one-sided
alternative PROB (<VAR>x</VAR> &gt; <VAR>y</VAR>) &gt; 1/2 is considered.  Similarly
for <CODE>&quot;&lt;&quot;</CODE>, the one-sided alternative PROB (<VAR>x</VAR> &gt; <VAR>y</VAR>) &lt;
1/2 is considered.  The default is the two-sided case.
</P>
<P>

The p-value of the test is returned in <VAR>pval</VAR>.
</P>
<P>

If no output argument is given, the p-value of the test is displayed.
</P>
</DL>
<P>

<A NAME="doc-z_test"></A>
<A NAME="IDX833"></A>
</P>
<DL>
<DT><U>Function File:</U> [<VAR>pval</VAR>, <VAR>z</VAR>] = <B>z_test</B> <I>(<VAR>x</VAR>, <VAR>m</VAR>, <VAR>v</VAR>, <VAR>alt</VAR>)</I>
<DD>Perform a Z-test of the null hypothesis <CODE>mean (<VAR>x</VAR>) ==
<VAR>m</VAR></CODE> for a sample <VAR>x</VAR> from a normal distribution with unknown
mean and known variance <VAR>v</VAR>.  Under the null, the test statistic
<VAR>z</VAR> follows a standard normal distribution.
<P>

With the optional argument string <VAR>alt</VAR>, the alternative of
interest can be selected.  If <VAR>alt</VAR> is <CODE>&quot;!=&quot;</CODE> or
<CODE>&quot;&lt;&gt;&quot;</CODE>, the null is tested against the two-sided alternative
<CODE>mean (<VAR>x</VAR>) != <VAR>m</VAR></CODE>.  If <VAR>alt</VAR> is <CODE>&quot;&gt;&quot;</CODE>, the
one-sided alternative <CODE>mean (<VAR>x</VAR>) &gt; <VAR>m</VAR></CODE> is considered.
Similarly for <CODE>&quot;&lt;&quot;</CODE>, the one-sided alternative <CODE>mean
(<VAR>x</VAR>) &lt; <VAR>m</VAR></CODE> is considered.  The default is the two-sided
case.
</P>
<P>

The p-value of the test is returned in <VAR>pval</VAR>.
</P>
<P>

If no output argument is given, the p-value of the test is displayed
along with some information.
</P>
</DL>
<P>

<A NAME="doc-z_test_2"></A>
<A NAME="IDX834"></A>
</P>
<DL>
<DT><U>Function File:</U> [<VAR>pval</VAR>, <VAR>z</VAR>] = <B>z_test_2</B> <I>(<VAR>x</VAR>, <VAR>y</VAR>, <VAR>v_x</VAR>, <VAR>v_y</VAR>, <VAR>alt</VAR>)</I>
<DD>For two samples <VAR>x</VAR> and <VAR>y</VAR> from normal distributions with
unknown means and known variances <VAR>v_x</VAR> and <VAR>v_y</VAR>, perform a
Z-test of the hypothesis of equal means.  Under the null, the test
statistic <VAR>z</VAR> follows a standard normal distribution.
<P>

With the optional argument string <VAR>alt</VAR>, the alternative of
interest can be selected.  If <VAR>alt</VAR> is <CODE>&quot;!=&quot;</CODE> or
<CODE>&quot;&lt;&gt;&quot;</CODE>, the null is tested against the two-sided alternative
<CODE>mean (<VAR>x</VAR>) != mean (<VAR>y</VAR>)</CODE>.  If alt is <CODE>&quot;&gt;&quot;</CODE>, the
one-sided alternative <CODE>mean (<VAR>x</VAR>) &gt; mean (<VAR>y</VAR>)</CODE> is used.
Similarly for <CODE>&quot;&lt;&quot;</CODE>, the one-sided alternative <CODE>mean
(<VAR>x</VAR>) &lt; mean (<VAR>y</VAR>)</CODE> is used.  The default is the two-sided
case.
</P>
<P>

The p-value of the test is returned in <VAR>pval</VAR>.
</P>
<P>

If no output argument is given, the p-value of the test is displayed
along with some information.
</P>
</DL>
<P>

<A NAME="Models"></A>
<HR SIZE="6">
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<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="octave_abt.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H2> 25.3 Models </H2>
<!--docid::SEC179::-->
<P>

<A NAME="doc-logistic_regression"></A>
<A NAME="IDX835"></A>
</P>
<DL>
<DT><U>Functio File:</U> [<VAR>theta</VAR>, <VAR>beta</VAR>, <VAR>dev</VAR>, <VAR>dl</VAR>, <VAR>d2l</VAR>, <VAR>p</VAR>] = <B>logistic_regression</B> <I>(<VAR>y</VAR>, <VAR>x</VAR>, <VAR>print</VAR>, <VAR>theta</VAR>, <VAR>beta</VAR>)</I>
<DD>Perform ordinal logistic regression.
<P>

Suppose <VAR>y</VAR> takes values in <VAR>k</VAR> ordered categories, and let
<CODE>gamma_i (<VAR>x</VAR>)</CODE> be the cumulative probability that <VAR>y</VAR>
falls in one of the first <VAR>i</VAR> categories given the covariate
<VAR>x</VAR>.  Then
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>[theta, beta] = logistic_regression (y, x)
</pre></td></tr></table><P>

fits the model
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>logit (gamma_i (x)) = theta_i - beta' * x,   i = 1, ..., k-1
</pre></td></tr></table><P>

The number of ordinal categories, <VAR>k</VAR>, is taken to be the number
of distinct values of <CODE>round (<VAR>y</VAR>)</CODE>.  If <VAR>k</VAR> equals 2,
<VAR>y</VAR> is binary and the model is ordinary logistic regression.  The
matrix <VAR>x</VAR> is assumed to have full column rank.
</P>
<P>

Given <VAR>y</VAR> only, <CODE>theta = logistic_regression (y)</CODE>
fits the model with baseline logit odds only.
</P>
<P>

The full form is
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>[theta, beta, dev, dl, d2l, gamma]
   = logistic_regression (y, x, print, theta, beta)
</pre></td></tr></table><P>

in which all output arguments and all input arguments except <VAR>y</VAR>
are optional.
</P>
<P>

Stting <VAR>print</VAR> to 1 requests summary information about the fitted
model to be displayed.  Setting <VAR>print</VAR> to 2 requests information
about convergence at each iteration.  Other values request no
information to be displayed.  The input arguments <VAR>theta</VAR> and
<VAR>beta</VAR> give initial estimates for <VAR>theta</VAR> and <VAR>beta</VAR>.
</P>
<P>

The returned value <VAR>dev</VAR> holds minus twice the log-likelihood.
</P>
<P>

The returned values <VAR>dl</VAR> and <VAR>d2l</VAR> are the vector of first
and the matrix of second derivatives of the log-likelihood with
respect to <VAR>theta</VAR> and <VAR>beta</VAR>.
</P>
<P>

<VAR>p</VAR> holds estimates for the conditional distribution of <VAR>y</VAR>
given <VAR>x</VAR>.
</P>
</DL>
<P>

<A NAME="Distributions"></A>
<HR SIZE="6">
<A NAME="SEC180"></A>
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<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="octave_toc.html#SEC_Contents">Contents</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="octave_42.html#SEC245">Index</A>]</TD>
<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="octave_abt.html#SEC_About"> ? </A>]</TD>
</TR></TABLE>
<H2> 25.4 Distributions </H2>
<!--docid::SEC180::-->
<P>

<A NAME="doc-beta_cdf"></A>
<A NAME="IDX836"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>beta_cdf</B> <I>(<VAR>x</VAR>, <VAR>a</VAR>, <VAR>b</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, returns the CDF at <VAR>x</VAR> of the beta
distribution with parameters <VAR>a</VAR> and <VAR>b</VAR>, i.e.,
PROB (beta (<VAR>a</VAR>, <VAR>b</VAR>) &lt;= <VAR>x</VAR>).
</DL>
<P>

<A NAME="doc-beta_inv"></A>
<A NAME="IDX837"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>beta_inv</B> <I>(<VAR>x</VAR>, <VAR>a</VAR>, <VAR>b</VAR>)</I>
<DD>For each component of <VAR>x</VAR>, compute the quantile (the inverse of
the CDF) at <VAR>x</VAR> of the Beta distribution with parameters <VAR>a</VAR>
and <VAR>b</VAR>.
</DL>
<P>

<A NAME="doc-beta_pdf"></A>
<A NAME="IDX838"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>beta_pdf</B> <I>(<VAR>x</VAR>, <VAR>a</VAR>, <VAR>b</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, returns the PDF at <VAR>x</VAR> of the beta
distribution with parameters <VAR>a</VAR> and <VAR>b</VAR>.
</DL>
<P>

<A NAME="doc-beta_rnd"></A>
<A NAME="IDX839"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>beta_rnd</B> <I>(<VAR>a</VAR>, <VAR>b</VAR>, <VAR>r</VAR>, <VAR>c</VAR>)</I>
<DD><A NAME="IDX840"></A>
<DT><U>Function File:</U>  <B>beta_rnd</B> <I>(<VAR>a</VAR>, <VAR>b</VAR>, <VAR>sz</VAR>)</I>
<DD>Return an <VAR>r</VAR> by <VAR>c</VAR> or <CODE>size (<VAR>sz</VAR>)</CODE> matrix of 
random samples from the Beta distribution with parameters <VAR>a</VAR> and
<VAR>b</VAR>.  Both <VAR>a</VAR> and <VAR>b</VAR> must be scalar or of size <VAR>r</VAR>
 by <VAR>c</VAR>.
<P>

If <VAR>r</VAR> and <VAR>c</VAR> are omitted, the size of the result matrix is
the common size of <VAR>a</VAR> and <VAR>b</VAR>.
</P>
</DL>
<P>

<A NAME="doc-binomial_cdf"></A>
<A NAME="IDX841"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>binomial_cdf</B> <I>(<VAR>x</VAR>, <VAR>n</VAR>, <VAR>p</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the CDF at <VAR>x</VAR> of the
binomial distribution with parameters <VAR>n</VAR> and <VAR>p</VAR>.
</DL>
<P>

<A NAME="doc-binomial_inv"></A>
<A NAME="IDX842"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>binomial_inv</B> <I>(<VAR>x</VAR>, <VAR>n</VAR>, <VAR>p</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the quantile at <VAR>x</VAR> of the
binomial distribution with parameters <VAR>n</VAR> and <VAR>p</VAR>.
</DL>
<P>

<A NAME="doc-binomial_pdf"></A>
<A NAME="IDX843"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>binomial_pdf</B> <I>(<VAR>x</VAR>, <VAR>n</VAR>, <VAR>p</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the probability density function
(PDF) at <VAR>x</VAR> of the binomial distribution with parameters <VAR>n</VAR>
and <VAR>p</VAR>.
</DL>
<P>

<A NAME="doc-binomial_rnd"></A>
<A NAME="IDX844"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>binomial_rnd</B> <I>(<VAR>n</VAR>, <VAR>p</VAR>, <VAR>r</VAR>, <VAR>c</VAR>)</I>
<DD><A NAME="IDX845"></A>
<DT><U>Function File:</U>  <B>binomial_rnd</B> <I>(<VAR>n</VAR>, <VAR>p</VAR>, <VAR>sz</VAR>)</I>
<DD>Return an <VAR>r</VAR> by <VAR>c</VAR>  or a <CODE>size (<VAR>sz</VAR>)</CODE> matrix of 
random samples from the binomial distribution with parameters <VAR>n</VAR>
and <VAR>p</VAR>.  Both <VAR>n</VAR> and <VAR>p</VAR> must be scalar or of size
<VAR>r</VAR> by <VAR>c</VAR>.
<P>

If <VAR>r</VAR> and <VAR>c</VAR> are omitted, the size of the result matrix is
the common size of <VAR>n</VAR> and <VAR>p</VAR>.
</P>
</DL>
<P>

<A NAME="doc-cauchy_cdf"></A>
<A NAME="IDX846"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>cauchy_cdf</B> <I>(<VAR>x</VAR>, <VAR>lambda</VAR>, <VAR>sigma</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the cumulative distribution
function (CDF) at <VAR>x</VAR> of the Cauchy distribution with location
parameter <VAR>lambda</VAR> and scale parameter <VAR>sigma</VAR>.  Default
values are <VAR>lambda</VAR> = 0, <VAR>sigma</VAR> = 1. 
</DL>
<P>

<A NAME="doc-cauchy_inv"></A>
<A NAME="IDX847"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>cauchy_inv</B> <I>(<VAR>x</VAR>, <VAR>lambda</VAR>, <VAR>sigma</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the quantile (the inverse of the
CDF) at <VAR>x</VAR> of the Cauchy distribution with location parameter
<VAR>lambda</VAR> and scale parameter <VAR>sigma</VAR>.  Default values are
<VAR>lambda</VAR> = 0, <VAR>sigma</VAR> = 1. 
</DL>
<P>

<A NAME="doc-cauchy_pdf"></A>
<A NAME="IDX848"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>cauchy_pdf</B> <I>(<VAR>x</VAR>, <VAR>lambda</VAR>, <VAR>sigma</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the probability density function
(PDF) at <VAR>x</VAR> of the Cauchy distribution with location parameter
<VAR>lambda</VAR> and scale parameter <VAR>sigma</VAR> &gt; 0.  Default values are
<VAR>lambda</VAR> = 0, <VAR>sigma</VAR> = 1. 
</DL>
<P>

<A NAME="doc-cauchy_rnd"></A>
<A NAME="IDX849"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>cauchy_rnd</B> <I>(<VAR>lambda</VAR>, <VAR>sigma</VAR>, <VAR>r</VAR>, <VAR>c</VAR>)</I>
<DD><A NAME="IDX850"></A>
<DT><U>Function File:</U>  <B>cauchy_rnd</B> <I>(<VAR>lambda</VAR>, <VAR>sigma</VAR>, <VAR>sz</VAR>)</I>
<DD>Return an <VAR>r</VAR> by <VAR>c</VAR> or a <CODE>size (<VAR>sz</VAR>)</CODE> matrix of 
random samples from the Cauchy distribution with parameters <VAR>lambda</VAR> 
and <VAR>sigma</VAR> which must both be scalar or of size <VAR>r</VAR> by <VAR>c</VAR>.
<P>

If <VAR>r</VAR> and <VAR>c</VAR> are omitted, the size of the result matrix is
the common size of <VAR>lambda</VAR> and <VAR>sigma</VAR>.
</P>
</DL>
<P>

<A NAME="doc-chisquare_cdf"></A>
<A NAME="IDX851"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>chisquare_cdf</B> <I>(<VAR>x</VAR>, <VAR>n</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the cumulative distribution
function (CDF) at <VAR>x</VAR> of the chisquare distribution with <VAR>n</VAR>
degrees of freedom.
</DL>
<P>

<A NAME="doc-chisquare_inv"></A>
<A NAME="IDX852"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>chisquare_inv</B> <I>(<VAR>x</VAR>, <VAR>n</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the quantile (the inverse of the
CDF) at <VAR>x</VAR> of the chisquare distribution with <VAR>n</VAR> degrees of
freedom.
</DL>
<P>

<A NAME="doc-chisquare_pdf"></A>
<A NAME="IDX853"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>chisquare_pdf</B> <I>(<VAR>x</VAR>, <VAR>n</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the probability density function
(PDF) at <VAR>x</VAR> of the chisquare distribution with <VAR>k</VAR> degrees
of freedom.
</DL>
<P>

<A NAME="doc-chisquare_rnd"></A>
<A NAME="IDX854"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>chisquare_rnd</B> <I>(<VAR>n</VAR>, <VAR>r</VAR>, <VAR>c</VAR>)</I>
<DD><A NAME="IDX855"></A>
<DT><U>Function File:</U>  <B>chisquare_rnd</B> <I>(<VAR>n</VAR>, <VAR>sz</VAR>)</I>
<DD>Return an <VAR>r</VAR> by <VAR>c</VAR>  or a <CODE>size (<VAR>sz</VAR>)</CODE> matrix of 
random samples from the chisquare distribution with <VAR>n</VAR> degrees 
of freedom.  <VAR>n</VAR> must be a scalar or of size <VAR>r</VAR> by <VAR>c</VAR>.
<P>

If <VAR>r</VAR> and <VAR>c</VAR> are omitted, the size of the result matrix is
the size of <VAR>n</VAR>.
</P>
</DL>
<P>

<A NAME="doc-discrete_cdf"></A>
<A NAME="IDX856"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>discrete_cdf</B> <I>(<VAR>x</VAR>, <VAR>v</VAR>, <VAR>p</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the cumulative distribution
function (CDF) at <VAR>x</VAR> of a univariate discrete distribution which
assumes the values in v with probabilities <VAR>p</VAR>.
</DL>
<P>

<A NAME="doc-discrete_inv"></A>
<A NAME="IDX857"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>discrete_inv</B> <I>(<VAR>x</VAR>, <VAR>v</VAR>, <VAR>p</VAR>)</I>
<DD>For each component of <VAR>x</VAR>, compute the quantile (the inverse of
the CDF) at <VAR>x</VAR> of the univariate distribution which assumes the
values in <VAR>v</VAR> with probabilities <VAR>p</VAR>.
</DL>
<P>

<A NAME="doc-discrete_pdf"></A>
<A NAME="IDX858"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>discrete_pdf</B> <I>(<VAR>x</VAR>, <VAR>v</VAR>, <VAR>p</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the probability density function
(pDF) at <VAR>x</VAR> of a univariate discrete distribution which assumes
the values in <VAR>v</VAR> with probabilities <VAR>p</VAR>.
</DL>
<P>

<A NAME="doc-discrete_rnd"></A>
<A NAME="IDX859"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>discrete_rnd</B> <I>(<VAR>n</VAR>, <VAR>v</VAR>, <VAR>p</VAR>)</I>
<DD><A NAME="IDX860"></A>
<DT><U>Function File:</U>  <B>discrete_rnd</B> <I>(<VAR>v</VAR>, <VAR>p</VAR>, <VAR>r</VAR>, <VAR>c</VAR>)</I>
<DD><A NAME="IDX861"></A>
<DT><U>Function File:</U>  <B>discrete_rnd</B> <I>(<VAR>v</VAR>, <VAR>p</VAR>, <VAR>sz</VAR>)</I>
<DD>Generate a row vector containing a random sample of size <VAR>n</VAR> from
the univariate distribution which assumes the values in <VAR>v</VAR> with
probabilities <VAR>p</VAR>. <VAR>n</VAR> must be a scalar.
<P>

If <VAR>r</VAR> and <VAR>c</VAR> are given create a matrix with <VAR>r</VAR> rows and
<VAR>c</VAR> columns. Or if <VAR>sz</VAR> is a vector, create a matrix of size
<VAR>sz</VAR>.
</P>
</DL>
<P>

<A NAME="doc-empirical_cdf"></A>
<A NAME="IDX862"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>empirical_cdf</B> <I>(<VAR>x</VAR>, <VAR>data</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the cumulative distribution
function (CDF) at <VAR>x</VAR> of the empirical distribution obtained from
the univariate sample <VAR>data</VAR>.
</DL>
<P>

<A NAME="doc-empirical_inv"></A>
<A NAME="IDX863"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>empirical_inv</B> <I>(<VAR>x</VAR>, <VAR>data</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the quantile (the inverse of the
CDF) at <VAR>x</VAR> of the empirical distribution obtained from the
univariate sample <VAR>data</VAR>.
</DL>
<P>

<A NAME="doc-empirical_pdf"></A>
<A NAME="IDX864"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>empirical_pdf</B> <I>(<VAR>x</VAR>, <VAR>data</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the probability density function
(PDF) at <VAR>x</VAR> of the empirical distribution obtained from the
univariate sample <VAR>data</VAR>.
</DL>
<P>

<A NAME="doc-empirical_rnd"></A>
<A NAME="IDX865"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>empirical_rnd</B> <I>(<VAR>n</VAR>, <VAR>data</VAR>)</I>
<DD><A NAME="IDX866"></A>
<DT><U>Function File:</U>  <B>empirical_rnd</B> <I>(<VAR>data</VAR>, <VAR>r</VAR>, <VAR>c</VAR>)</I>
<DD><A NAME="IDX867"></A>
<DT><U>Function File:</U>  <B>empirical_rnd</B> <I>(<VAR>data</VAR>, <VAR>sz</VAR>)</I>
<DD>Generate a bootstrap sample of size <VAR>n</VAR> from the empirical
distribution obtained from the univariate sample <VAR>data</VAR>.
<P>

If <VAR>r</VAR> and <VAR>c</VAR> are given create a matrix with <VAR>r</VAR> rows and
<VAR>c</VAR> columns. Or if <VAR>sz</VAR> is a vector, create a matrix of size
<VAR>sz</VAR>.
</P>
</DL>
<P>

<A NAME="doc-exponential_cdf"></A>
<A NAME="IDX868"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>exponential_cdf</B> <I>(<VAR>x</VAR>, <VAR>lambda</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the cumulative distribution
function (CDF) at <VAR>x</VAR> of the exponential distribution with
parameter <VAR>lambda</VAR>.
<P>

The arguments can be of common size or scalar.
</P>
</DL>
<P>

<A NAME="doc-exponential_inv"></A>
<A NAME="IDX869"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>exponential_inv</B> <I>(<VAR>x</VAR>, <VAR>lambda</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the quantile (the inverse of the
CDF) at <VAR>x</VAR> of the exponential distribution with parameter
<VAR>lambda</VAR>.
</DL>
<P>

<A NAME="doc-exponential_pdf"></A>
<A NAME="IDX870"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>exponential_pdf</B> <I>(<VAR>x</VAR>, <VAR>lambda</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the probability density function
(PDF) of the exponential distribution with parameter <VAR>lambda</VAR>.
</DL>
<P>

<A NAME="doc-exponential_rnd"></A>
<A NAME="IDX871"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>exponential_rnd</B> <I>(<VAR>lambda</VAR>, <VAR>r</VAR>, <VAR>c</VAR>)</I>
<DD><A NAME="IDX872"></A>
<DT><U>Function File:</U>  <B>exponential_rnd</B> <I>(<VAR>lambda</VAR>, <VAR>sz</VAR>)</I>
<DD>Return an <VAR>r</VAR> by <VAR>c</VAR> matrix of random samples from the
exponential distribution with parameter <VAR>lambda</VAR>, which must be a
scalar or of size <VAR>r</VAR> by <VAR>c</VAR>. Or if <VAR>sz</VAR> is a vector, 
create a matrix of size <VAR>sz</VAR>.
<P>

If <VAR>r</VAR> and <VAR>c</VAR> are omitted, the size of the result matrix is
the size of <VAR>lambda</VAR>.
</P>
</DL>
<P>

<A NAME="doc-f_cdf"></A>
<A NAME="IDX873"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>f_cdf</B> <I>(<VAR>x</VAR>, <VAR>m</VAR>, <VAR>n</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the CDF at <VAR>x</VAR> of the F
distribution with <VAR>m</VAR> and <VAR>n</VAR> degrees of freedom, i.e.,
PROB (F (<VAR>m</VAR>, <VAR>n</VAR>) &lt;= <VAR>x</VAR>). 
</DL>
<P>

<A NAME="doc-f_inv"></A>
<A NAME="IDX874"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>f_inv</B> <I>(<VAR>x</VAR>, <VAR>m</VAR>, <VAR>n</VAR>)</I>
<DD>For each component of <VAR>x</VAR>, compute the quantile (the inverse of
the CDF) at <VAR>x</VAR> of the F distribution with parameters <VAR>m</VAR> and
<VAR>n</VAR>.
</DL>
<P>

<A NAME="doc-f_pdf"></A>
<A NAME="IDX875"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>f_pdf</B> <I>(<VAR>x</VAR>, <VAR>m</VAR>, <VAR>n</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the probability density function
(PDF) at <VAR>x</VAR> of the F distribution with <VAR>m</VAR> and <VAR>n</VAR>
degrees of freedom.
</DL>
<P>

<A NAME="doc-f_rnd"></A>
<A NAME="IDX876"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>f_rnd</B> <I>(<VAR>m</VAR>, <VAR>n</VAR>, <VAR>r</VAR>, <VAR>c</VAR>)</I>
<DD><A NAME="IDX877"></A>
<DT><U>Function File:</U>  <B>f_rnd</B> <I>(<VAR>m</VAR>, <VAR>n</VAR>, <VAR>sz</VAR>)</I>
<DD>Return an <VAR>r</VAR> by <VAR>c</VAR> matrix of random samples from the F
distribution with <VAR>m</VAR> and <VAR>n</VAR> degrees of freedom.  Both
<VAR>m</VAR> and <VAR>n</VAR> must be scalar or of size <VAR>r</VAR> by <VAR>c</VAR>.
If <VAR>sz</VAR> is a vector the random samples are in a matrix of 
size <VAR>sz</VAR>.
<P>

If <VAR>r</VAR> and <VAR>c</VAR> are omitted, the size of the result matrix is
the common size of <VAR>m</VAR> and <VAR>n</VAR>.
</P>
</DL>
<P>

<A NAME="doc-gamma_cdf"></A>
<A NAME="IDX878"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>gamma_cdf</B> <I>(<VAR>x</VAR>, <VAR>a</VAR>, <VAR>b</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the cumulative distribution
function (CDF) at <VAR>x</VAR> of the Gamma distribution with parameters
<VAR>a</VAR> and <VAR>b</VAR>.
</DL>
<P>

<A NAME="doc-gamma_inv"></A>
<A NAME="IDX879"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>gamma_inv</B> <I>(<VAR>x</VAR>, <VAR>a</VAR>, <VAR>b</VAR>)</I>
<DD>For each component of <VAR>x</VAR>, compute the quantile (the inverse of
the CDF) at <VAR>x</VAR> of the Gamma distribution with parameters <VAR>a</VAR>
and <VAR>b</VAR>. 
</DL>
<P>

<A NAME="doc-gamma_pdf"></A>
<A NAME="IDX880"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>gamma_pdf</B> <I>(<VAR>x</VAR>, <VAR>a</VAR>, <VAR>b</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, return the probability density function
(PDF) at <VAR>x</VAR> of the Gamma distribution with parameters <VAR>a</VAR>
and <VAR>b</VAR>.
</DL>
<P>

<A NAME="doc-gamma_rnd"></A>
<A NAME="IDX881"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>gamma_rnd</B> <I>(<VAR>a</VAR>, <VAR>b</VAR>, <VAR>r</VAR>, <VAR>c</VAR>)</I>
<DD><A NAME="IDX882"></A>
<DT><U>Function File:</U>  <B>gamma_rnd</B> <I>(<VAR>a</VAR>, <VAR>b</VAR>, <VAR>sz</VAR>)</I>
<DD>Return an <VAR>r</VAR> by <VAR>c</VAR> or a <CODE>size (<VAR>sz</VAR>)</CODE> matrix of 
random samples from the Gamma distribution with parameters <VAR>a</VAR>
and <VAR>b</VAR>.  Both <VAR>a</VAR> and <VAR>b</VAR> must be scalar or of size 
<VAR>r</VAR> by <VAR>c</VAR>.
<P>

If <VAR>r</VAR> and <VAR>c</VAR> are omitted, the size of the result matrix is
the common size of <VAR>a</VAR> and <VAR>b</VAR>.
</P>
</DL>
<P>

<A NAME="doc-geometric_cdf"></A>
<A NAME="IDX883"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>geometric_cdf</B> <I>(<VAR>x</VAR>, <VAR>p</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the CDF at <VAR>x</VAR> of the
geometric distribution with parameter <VAR>p</VAR>.
</DL>
<P>

<A NAME="doc-geometric_inv"></A>
<A NAME="IDX884"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>geometric_inv</B> <I>(<VAR>x</VAR>, <VAR>p</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the quantile at <VAR>x</VAR> of the
geometric distribution with parameter <VAR>p</VAR>.
</DL>
<P>

<A NAME="doc-geometric_pdf"></A>
<A NAME="IDX885"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>geometric_pdf</B> <I>(<VAR>x</VAR>, <VAR>p</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the probability density function
(PDF) at <VAR>x</VAR> of the geometric distribution with parameter <VAR>p</VAR>.
</DL>
<P>

<A NAME="doc-geometric_rnd"></A>
<A NAME="IDX886"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>geometric_rnd</B> <I>(<VAR>p</VAR>, <VAR>r</VAR>, <VAR>c</VAR>)</I>
<DD><A NAME="IDX887"></A>
<DT><U>Function File:</U>  <B>geometric_rnd</B> <I>(<VAR>p</VAR>, <VAR>sz</VAR>)</I>
<DD>Return an <VAR>r</VAR> by <VAR>c</VAR> matrix of random samples from the
geometric distribution with parameter <VAR>p</VAR>, which must be a scalar
or of size <VAR>r</VAR> by <VAR>c</VAR>.
<P>

If <VAR>r</VAR> and <VAR>c</VAR> are given create a matrix with <VAR>r</VAR> rows and
<VAR>c</VAR> columns. Or if <VAR>sz</VAR> is a vector, create a matrix of size
<VAR>sz</VAR>.
</P>
</DL>
<P>

<A NAME="doc-hypergeometric_cdf"></A>
<A NAME="IDX888"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>hypergeometric_cdf</B> <I>(<VAR>x</VAR>, <VAR>m</VAR>, <VAR>t</VAR>, <VAR>n</VAR>)</I>
<DD>Compute the cumulative distribution function (CDF) at <VAR>x</VAR> of the
hypergeometric distribution with parameters <VAR>m</VAR>, <VAR>t</VAR>, and
<VAR>n</VAR>.  This is the probability of obtaining not more than <VAR>x</VAR>
marked items when randomly drawing a sample of size <VAR>n</VAR> without
replacement from a population of total size <VAR>t</VAR> containing
<VAR>m</VAR> marked items.
<P>

The parameters <VAR>m</VAR>, <VAR>t</VAR>, and <VAR>n</VAR> must positive integers
with <VAR>m</VAR> and <VAR>n</VAR> not greater than <VAR>t</VAR>.
</P>
</DL>
<P>

<A NAME="doc-hypergeometric_inv"></A>
<A NAME="IDX889"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>hypergeometric_inv</B> <I>(<VAR>x</VAR>, <VAR>m</VAR>, <VAR>t</VAR>, <VAR>n</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the quantile at <VAR>x</VAR> of the
hypergeometric distribution with parameters <VAR>m</VAR>, <VAR>t</VAR>, and
<VAR>n</VAR>.
<P>

The parameters <VAR>m</VAR>, <VAR>t</VAR>, and <VAR>n</VAR> must positive integers
with <VAR>m</VAR> and <VAR>n</VAR> not greater than <VAR>t</VAR>.
</P>
</DL>
<P>

<A NAME="doc-hypergeometric_pdf"></A>
<A NAME="IDX890"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>hypergeometric_pdf</B> <I>(<VAR>x</VAR>, <VAR>m</VAR>, <VAR>t</VAR>, <VAR>n</VAR>)</I>
<DD>Compute the probability density function (PDF) at <VAR>x</VAR> of the
hypergeometric distribution with parameters <VAR>m</VAR>, <VAR>t</VAR>, and
<VAR>n</VAR>. This is the probability of obtaining <VAR>x</VAR> marked items
when randomly drawing a sample of size <VAR>n</VAR> without replacement
from a population of total size <VAR>t</VAR> containing <VAR>m</VAR> marked items.
<P>

The arguments must be of common size or scalar.
</P>
</DL>
<P>

<A NAME="doc-hypergeometric_rnd"></A>
<A NAME="IDX891"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>hypergeometric_rnd</B> <I>(<VAR>n_size</VAR>, <VAR>m</VAR>, <VAR>t</VAR>, <VAR>n</VAR>)</I>
<DD><A NAME="IDX892"></A>
<DT><U>Function File:</U>  <B>hypergeometric_rnd</B> <I>(<VAR>m</VAR>, <VAR>t</VAR>, <VAR>n</VAR>, <VAR>r</VAR>, <VAR>c</VAR>)</I>
<DD><A NAME="IDX893"></A>
<DT><U>Function File:</U>  <B>hypergeometric_rnd</B> <I>(<VAR>m</VAR>, <VAR>t</VAR>, <VAR>n</VAR>, <VAR>sz</VAR>)</I>
<DD>Generate a row vector containing a random sample of size <VAR>n_size</VAR>
from the hypergeometric distribution with parameters <VAR>m</VAR>, <VAR>t</VAR>,
and <VAR>n</VAR>.
<P>

If  <VAR>r</VAR> and <VAR>c</VAR> are given create a matrix with <VAR>r</VAR> rows and
<VAR>c</VAR> columns. Or if <VAR>sz</VAR> is a vector, create a matrix of size
<VAR>sz</VAR>.
</P>
<P>

The parameters <VAR>m</VAR>, <VAR>t</VAR>, and <VAR>n</VAR> must positive integers
with <VAR>m</VAR> and <VAR>n</VAR> not greater than <VAR>t</VAR>.
</P>
</DL>
<P>

<A NAME="doc-kolmogorov_smirnov_cdf"></A>
<A NAME="IDX894"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>kolmogorov_smirnov_cdf</B> <I>(<VAR>x</VAR>, <VAR>tol</VAR>)</I>
<DD>Return the CDF at <VAR>x</VAR> of the Kolmogorov-Smirnov distribution,
<TABLE><tr><td>&nbsp;</td><td class=example><pre>         Inf
Q(x) =   SUM    (-1)^k exp(-2 k^2 x^2)
       k = -Inf
</pre></td></tr></table><P>

for <VAR>x</VAR> &gt; 0.
</P>
<P>

The optional parameter <VAR>tol</VAR> specifies the precision up to which
the series should be evaluated;  the default is <VAR>tol</VAR> = <CODE>eps</CODE>.
</P>
</DL>
<P>

<A NAME="doc-laplace_cdf"></A>
<A NAME="IDX895"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>laplace_cdf</B> <I>(<VAR>x</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the cumulative distribution
function (CDF) at <VAR>x</VAR> of the Laplace distribution.
</DL>
<P>

<A NAME="doc-laplace_inv"></A>
<A NAME="IDX896"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>laplace_inv</B> <I>(<VAR>x</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the quantile (the inverse of the
CDF) at <VAR>x</VAR> of the Laplace distribution.
</DL>
<P>

<A NAME="doc-laplace_pdf"></A>
<A NAME="IDX897"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>laplace_pdf</B> <I>(<VAR>x</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the probability density function
(PDF) at <VAR>x</VAR> of the Laplace distribution.
</DL>
<P>

<A NAME="doc-laplace_rnd"></A>
<A NAME="IDX898"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>laplace_rnd</B> <I>(<VAR>r</VAR>, <VAR>c</VAR>)</I>
<DD><A NAME="IDX899"></A>
<DT><U>Function File:</U>  <B>laplace_rnd</B> <I>(<VAR>sz</VAR>);</I>
<DD>Return an <VAR>r</VAR> by <VAR>c</VAR> matrix of random numbers from the
Laplace distribution. Or is <VAR>sz</VAR> is a vector, create a matrix of
<VAR>sz</VAR>.
</DL>
<P>

<A NAME="doc-logistic_cdf"></A>
<A NAME="IDX900"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>logistic_cdf</B> <I>(<VAR>x</VAR>)</I>
<DD>For each component of <VAR>x</VAR>, compute the CDF at <VAR>x</VAR> of the
logistic distribution.
</DL>
<P>

<A NAME="doc-logistic_inv"></A>
<A NAME="IDX901"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>logistic_inv</B> <I>(<VAR>x</VAR>)</I>
<DD>For each component of <VAR>x</VAR>, compute the quantile (the inverse of
the CDF) at <VAR>x</VAR> of the logistic distribution.
</DL>
<P>

<A NAME="doc-logistic_pdf"></A>
<A NAME="IDX902"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>logistic_pdf</B> <I>(<VAR>x</VAR>)</I>
<DD>For each component of <VAR>x</VAR>, compute the PDF at <VAR>x</VAR> of the
logistic distribution.
</DL>
<P>

<A NAME="doc-logistic_rnd"></A>
<A NAME="IDX903"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>logistic_rnd</B> <I>(<VAR>r</VAR>, <VAR>c</VAR>)</I>
<DD><A NAME="IDX904"></A>
<DT><U>Function File:</U>  <B>logistic_rnd</B> <I>(<VAR>sz</VAR>)</I>
<DD>Return an <VAR>r</VAR> by <VAR>c</VAR> matrix of random numbers from the
logistic distribution. Or is <VAR>sz</VAR> is a vector, create a matrix of
<VAR>sz</VAR>.
</DL>
<P>

<A NAME="doc-lognormal_cdf"></A>
<A NAME="IDX905"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>lognormal_cdf</B> <I>(<VAR>x</VAR>, <VAR>a</VAR>, <VAR>v</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the cumulative distribution
function (CDF) at <VAR>x</VAR> of the lognormal distribution with
parameters <VAR>a</VAR> and <VAR>v</VAR>.  If a random variable follows this
distribution, its logarithm is normally distributed with mean
<CODE>log (<VAR>a</VAR>)</CODE> and variance <VAR>v</VAR>.
<P>

Default values are <VAR>a</VAR> = 1, <VAR>v</VAR> = 1.
</P>
</DL>
<P>

<A NAME="doc-lognormal_inv"></A>
<A NAME="IDX906"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>lognormal_inv</B> <I>(<VAR>x</VAR>, <VAR>a</VAR>, <VAR>v</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the quantile (the inverse of the
CDF) at <VAR>x</VAR> of the lognormal distribution with parameters <VAR>a</VAR>
and <VAR>v</VAR>.  If a random variable follows this distribution, its
logarithm is normally distributed with mean <CODE>log (<VAR>a</VAR>)</CODE> and
variance <VAR>v</VAR>.
<P>

Default values are <VAR>a</VAR> = 1, <VAR>v</VAR> = 1.
</P>
</DL>
<P>

<A NAME="doc-lognormal_pdf"></A>
<A NAME="IDX907"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>lognormal_pdf</B> <I>(<VAR>x</VAR>, <VAR>a</VAR>, <VAR>v</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the probability density function
(PDF) at <VAR>x</VAR> of the lognormal distribution with parameters
<VAR>a</VAR> and <VAR>v</VAR>.  If a random variable follows this distribution,
its logarithm is normally distributed with mean <CODE>log (<VAR>a</VAR>)</CODE>
and variance <VAR>v</VAR>.
<P>

Default values are <VAR>a</VAR> = 1, <VAR>v</VAR> = 1.
</P>
</DL>
<P>

<A NAME="doc-lognormal_rnd"></A>
<A NAME="IDX908"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>lognormal_rnd</B> <I>(<VAR>a</VAR>, <VAR>v</VAR>, <VAR>r</VAR>, <VAR>c</VAR>)</I>
<DD><A NAME="IDX909"></A>
<DT><U>Function File:</U>  <B>lognormal_rnd</B> <I>(<VAR>a</VAR>, <VAR>v</VAR>, <VAR>sz</VAR>)</I>
<DD>Return an <VAR>r</VAR> by <VAR>c</VAR> matrix of random samples from the
lognormal distribution with parameters <VAR>a</VAR> and <VAR>v</VAR>. Both
<VAR>a</VAR> and <VAR>v</VAR> must be scalar or of size <VAR>r</VAR> by <VAR>c</VAR>.
Or if <VAR>sz</VAR> is a vector, create a matrix of size <VAR>sz</VAR>.
<P>

If <VAR>r</VAR> and <VAR>c</VAR> are omitted, the size of the result matrix is
the common size of <VAR>a</VAR> and <VAR>v</VAR>.
</P>
</DL>
<P>

<A NAME="doc-normal_cdf"></A>
<A NAME="IDX910"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>normal_cdf</B> <I>(<VAR>x</VAR>, <VAR>m</VAR>, <VAR>v</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the cumulative distribution
function (CDF) at <VAR>x</VAR> of the normal distribution with mean
<VAR>m</VAR> and variance <VAR>v</VAR>.
<P>

Default values are <VAR>m</VAR> = 0, <VAR>v</VAR> = 1.
</P>
</DL>
<P>

<A NAME="doc-normal_inv"></A>
<A NAME="IDX911"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>normal_inv</B> <I>(<VAR>x</VAR>, <VAR>m</VAR>, <VAR>v</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the quantile (the inverse of the
CDF) at <VAR>x</VAR> of the normal distribution with mean <VAR>m</VAR> and
variance <VAR>v</VAR>.
<P>

Default values are <VAR>m</VAR> = 0, <VAR>v</VAR> = 1.
</P>
</DL>
<P>

<A NAME="doc-normal_pdf"></A>
<A NAME="IDX912"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>normal_pdf</B> <I>(<VAR>x</VAR>, <VAR>m</VAR>, <VAR>v</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the probability density function
(PDF) at <VAR>x</VAR> of the normal distribution with mean <VAR>m</VAR> and
variance <VAR>v</VAR>.
<P>

Default values are <VAR>m</VAR> = 0, <VAR>v</VAR> = 1.
</P>
</DL>
<P>

<A NAME="doc-normal_rnd"></A>
<A NAME="IDX913"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>normal_rnd</B> <I>(<VAR>m</VAR>, <VAR>v</VAR>, <VAR>r</VAR>, <VAR>c</VAR>)</I>
<DD><A NAME="IDX914"></A>
<DT><U>Function File:</U>  <B>normal_rnd</B> <I>(<VAR>m</VAR>, <VAR>v</VAR>, <VAR>sz</VAR>)</I>
<DD>Return an <VAR>r</VAR> by <VAR>c</VAR>  or <CODE>size (<VAR>sz</VAR>)</CODE> matrix of
random samples from the normal distribution with parameters <VAR>m</VAR> 
and <VAR>v</VAR>.  Both <VAR>m</VAR> and <VAR>v</VAR> must be scalar or of size 
<VAR>r</VAR> by <VAR>c</VAR>.
<P>

If <VAR>r</VAR> and <VAR>c</VAR> are omitted, the size of the result matrix is
the common size of <VAR>m</VAR> and <VAR>v</VAR>.
</P>
</DL>
<P>

<A NAME="doc-pascal_cdf"></A>
<A NAME="IDX915"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>pascal_cdf</B> <I>(<VAR>x</VAR>, <VAR>n</VAR>, <VAR>p</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the CDF at x of the Pascal
(negative binomial) distribution with parameters <VAR>n</VAR> and <VAR>p</VAR>.
<P>

The number of failures in a Bernoulli experiment with success
probability <VAR>p</VAR> before the <VAR>n</VAR>-th success follows this
distribution.
</P>
</DL>
<P>

<A NAME="doc-pascal_inv"></A>
<A NAME="IDX916"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>pascal_inv</B> <I>(<VAR>x</VAR>, <VAR>n</VAR>, <VAR>p</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the quantile at <VAR>x</VAR> of the
Pascal (negative binomial) distribution with parameters <VAR>n</VAR> and
<VAR>p</VAR>.
<P>

The number of failures in a Bernoulli experiment with success
probability <VAR>p</VAR> before the <VAR>n</VAR>-th success follows this
distribution.
</P>
</DL>
<P>

<A NAME="doc-pascal_pdf"></A>
<A NAME="IDX917"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>pascal_pdf</B> <I>(<VAR>x</VAR>, <VAR>n</VAR>, <VAR>p</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the probability density function
(PDF) at <VAR>x</VAR> of the Pascal (negative binomial) distribution with
parameters <VAR>n</VAR> and <VAR>p</VAR>.
<P>

The number of failures in a Bernoulli experiment with success
probability <VAR>p</VAR> before the <VAR>n</VAR>-th success follows this
distribution. 
</P>
</DL>
<P>

<A NAME="doc-pascal_rnd"></A>
<A NAME="IDX918"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>pascal_rnd</B> <I>(<VAR>n</VAR>, <VAR>p</VAR>, <VAR>r</VAR>, <VAR>c</VAR>)</I>
<DD><A NAME="IDX919"></A>
<DT><U>Function File:</U>  <B>pascal_rnd</B> <I>(<VAR>n</VAR>, <VAR>p</VAR>, <VAR>sz</VAR>)</I>
<DD>Return an <VAR>r</VAR> by <VAR>c</VAR> matrix of random samples from the Pascal
(negative binomial) distribution with parameters <VAR>n</VAR> and <VAR>p</VAR>.
Both <VAR>n</VAR> and <VAR>p</VAR> must be scalar or of size <VAR>r</VAR> by <VAR>c</VAR>.
<P>

If <VAR>r</VAR> and <VAR>c</VAR> are omitted, the size of the result matrix is
the common size of <VAR>n</VAR> and <VAR>p</VAR>. Or if <VAR>sz</VAR> is a vector, 
create a matrix of size <VAR>sz</VAR>.
</P>
</DL>
<P>

<A NAME="doc-poisson_cdf"></A>
<A NAME="IDX920"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>poisson_cdf</B> <I>(<VAR>x</VAR>, <VAR>lambda</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the cumulative distribution
function (CDF) at <VAR>x</VAR> of the Poisson distribution with parameter
lambda.
</DL>
<P>

<A NAME="doc-poisson_inv"></A>
<A NAME="IDX921"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>poisson_inv</B> <I>(<VAR>x</VAR>, <VAR>lambda</VAR>)</I>
<DD>For each component of <VAR>x</VAR>, compute the quantile (the inverse of
the CDF) at <VAR>x</VAR> of the Poisson distribution with parameter
<VAR>lambda</VAR>.
</DL>
<P>

<A NAME="doc-poisson_pdf"></A>
<A NAME="IDX922"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>poisson_pdf</B> <I>(<VAR>x</VAR>, <VAR>lambda</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the probability density function
(PDF) at <VAR>x</VAR> of the poisson distribution with parameter <VAR>lambda</VAR>.
</DL>
<P>

<A NAME="doc-poisson_rnd"></A>
<A NAME="IDX923"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>poisson_rnd</B> <I>(<VAR>lambda</VAR>, <VAR>r</VAR>, <VAR>c</VAR>)</I>
<DD>Return an <VAR>r</VAR> by <VAR>c</VAR> matrix of random samples from the
Poisson distribution with parameter <VAR>lambda</VAR>, which must be a 
scalar or of size <VAR>r</VAR> by <VAR>c</VAR>.
<P>

If <VAR>r</VAR> and <VAR>c</VAR> are omitted, the size of the result matrix is
the size of <VAR>lambda</VAR>.
</P>
</DL>
<P>

<A NAME="doc-stdnormal_cdf"></A>
<A NAME="IDX924"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>stdnormal_cdf</B> <I>(<VAR>x</VAR>)</I>
<DD>For each component of <VAR>x</VAR>, compute the CDF of the standard normal
distribution at <VAR>x</VAR>.
</DL>
<P>

<A NAME="doc-stdnormal_inv"></A>
<A NAME="IDX925"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>stdnormal_inv</B> <I>(<VAR>x</VAR>)</I>
<DD>For each component of <VAR>x</VAR>, compute compute the quantile (the
inverse of the CDF) at <VAR>x</VAR> of the standard normal distribution.
</DL>
<P>

<A NAME="doc-stdnormal_pdf"></A>
<A NAME="IDX926"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>stdnormal_pdf</B> <I>(<VAR>x</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the probability density function
(PDF) of the standard normal distribution at <VAR>x</VAR>.
</DL>
<P>

<A NAME="doc-stdnormal_rnd"></A>
<A NAME="IDX927"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>stdnormal_rnd</B> <I>(<VAR>r</VAR>, <VAR>c</VAR>)</I>
<DD><A NAME="IDX928"></A>
<DT><U>Function File:</U>  <B>stdnormal_rnd</B> <I>(<VAR>sz</VAR>)</I>
<DD>Return an <VAR>r</VAR> by <VAR>c</VAR> or <CODE>size (<VAR>sz</VAR>)</CODE> matrix of 
random numbers from the standard normal distribution.
</DL>
<P>

<A NAME="doc-t_cdf"></A>
<A NAME="IDX929"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>t_cdf</B> <I>(<VAR>x</VAR>, <VAR>n</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the CDF at <VAR>x</VAR> of the
t (Student) distribution with <VAR>n</VAR> degrees of freedom, i.e.,
PROB (t(<VAR>n</VAR>) &lt;= <VAR>x</VAR>).
</DL>
<P>

<A NAME="doc-t_inv"></A>
<A NAME="IDX930"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>t_inv</B> <I>(<VAR>x</VAR>, <VAR>n</VAR>)</I>
<DD>For each component of <VAR>x</VAR>, compute the quantile (the inverse of
the CDF) at <VAR>x</VAR> of the t (Student) distribution with parameter
<VAR>n</VAR>.
</DL>
<P>

<A NAME="doc-t_pdf"></A>
<A NAME="IDX931"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>t_pdf</B> <I>(<VAR>x</VAR>, <VAR>n</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the probability density function
(PDF) at <VAR>x</VAR> of the <VAR>t</VAR> (Student) distribution with <VAR>n</VAR>
degrees of freedom. 
</DL>
<P>

<A NAME="doc-t_rnd"></A>
<A NAME="IDX932"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>t_rnd</B> <I>(<VAR>n</VAR>, <VAR>r</VAR>, <VAR>c</VAR>)</I>
<DD><A NAME="IDX933"></A>
<DT><U>Function File:</U>  <B>t_rnd</B> <I>(<VAR>n</VAR>, <VAR>sz</VAR>)</I>
<DD>Return an <VAR>r</VAR> by <VAR>c</VAR> matrix of random samples from the t
(Student) distribution with <VAR>n</VAR> degrees of freedom.  <VAR>n</VAR> must
be a scalar or of size <VAR>r</VAR> by <VAR>c</VAR>. Or if <VAR>sz</VAR> is a
vector create a matrix of size <VAR>sz</VAR>.
<P>

If <VAR>r</VAR> and <VAR>c</VAR> are omitted, the size of the result matrix is
the size of <VAR>n</VAR>.
</P>
</DL>
<P>

<A NAME="doc-uniform_cdf"></A>
<A NAME="IDX934"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>uniform_cdf</B> <I>(<VAR>x</VAR>, <VAR>a</VAR>, <VAR>b</VAR>)</I>
<DD>Return the CDF at <VAR>x</VAR> of the uniform distribution on [<VAR>a</VAR>,
<VAR>b</VAR>], i.e., PROB (uniform (<VAR>a</VAR>, <VAR>b</VAR>) &lt;= x).
<P>

Default values are <VAR>a</VAR> = 0, <VAR>b</VAR> = 1.
</P>
</DL>
<P>

<A NAME="doc-uniform_inv"></A>
<A NAME="IDX935"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>uniform_inv</B> <I>(<VAR>x</VAR>, <VAR>a</VAR>, <VAR>b</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the quantile (the inverse of the
CDF) at <VAR>x</VAR> of the uniform distribution on [<VAR>a</VAR>, <VAR>b</VAR>].
<P>

Default values are <VAR>a</VAR> = 0, <VAR>b</VAR> = 1.
</P>
</DL>
<P>

<A NAME="doc-uniform_pdf"></A>
<A NAME="IDX936"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>uniform_pdf</B> <I>(<VAR>x</VAR>, <VAR>a</VAR>, <VAR>b</VAR>)</I>
<DD>For each element of <VAR>x</VAR>, compute the PDF at <VAR>x</VAR> of the uniform
distribution on [<VAR>a</VAR>, <VAR>b</VAR>].
<P>

Default values are <VAR>a</VAR> = 0, <VAR>b</VAR> = 1.
</P>
</DL>
<P>

<A NAME="doc-uniform_rnd"></A>
<A NAME="IDX937"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>uniform_rnd</B> <I>(<VAR>a</VAR>, <VAR>b</VAR>, <VAR>r</VAR>, <VAR>c</VAR>)</I>
<DD><A NAME="IDX938"></A>
<DT><U>Function File:</U>  <B>uniform_rnd</B> <I>(<VAR>a</VAR>, <VAR>b</VAR>, <VAR>sz</VAR>)</I>
<DD>Return an <VAR>r</VAR> by <VAR>c</VAR> or a <CODE>size (<VAR>sz</VAR>)</CODE> matrix of 
random samples from the uniform distribution on [<VAR>a</VAR>, <VAR>b</VAR>]. 
Both <VAR>a</VAR> and <VAR>b</VAR> must be scalar or of size <VAR>r</VAR> by <VAR>c</VAR>.
<P>

If <VAR>r</VAR> and <VAR>c</VAR> are omitted, the size of the result matrix is
the common size of <VAR>a</VAR> and <VAR>b</VAR>.
</P>
</DL>
<P>

<A NAME="doc-weibull_cdf"></A>
<A NAME="IDX939"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>weibull_cdf</B> <I>(<VAR>x</VAR>, <VAR>alpha</VAR>, <VAR>sigma</VAR>)</I>
<DD>Compute the cumulative distribution function (CDF) at <VAR>x</VAR> of the
Weibull distribution with shape parameter <VAR>alpha</VAR> and scale
parameter <VAR>sigma</VAR>, which is
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>1 - exp(-(x/sigma)^alpha)
</pre></td></tr></table><P>

for <VAR>x</VAR> &gt;= 0.
</P>
</DL>
<P>

<A NAME="doc-weibull_inv"></A>
<A NAME="IDX940"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>weibull_inv</B> <I>(<VAR>x</VAR>, <VAR>lambda</VAR>, <VAR>alpha</VAR>)</I>
<DD>Compute the quantile (the inverse of the CDF) at <VAR>x</VAR> of the
Weibull distribution with shape parameter <VAR>alpha</VAR> and scale
parameter <VAR>sigma</VAR>.
</DL>
<P>

<A NAME="doc-weibull_pdf"></A>
<A NAME="IDX941"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>weibull_pdf</B> <I>(<VAR>x</VAR>, <VAR>alpha</VAR>, <VAR>sigma</VAR>)</I>
<DD>Compute the probability density function (PDF) at <VAR>x</VAR> of the
Weibull distribution with shape parameter <VAR>alpha</VAR> and scale
parameter <VAR>sigma</VAR> which is given by
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>   alpha * sigma^(-alpha) * x^(alpha-1) * exp(-(x/sigma)^alpha)
</pre></td></tr></table><P>

for <VAR>x</VAR> &gt; 0.
</P>
</DL>
<P>

<A NAME="doc-weibull_rnd"></A>
<A NAME="IDX942"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>weibull_rnd</B> <I>(<VAR>alpha</VAR>, <VAR>sigma</VAR>, <VAR>r</VAR>, <VAR>c</VAR>)</I>
<DD><A NAME="IDX943"></A>
<DT><U>Function File:</U>  <B>weibull_rnd</B> <I>(<VAR>alpha</VAR>, <VAR>sigma</VAR>, <VAR>sz</VAR>)</I>
<DD>Return an <VAR>r</VAR> by <VAR>c</VAR> matrix of random samples from the
Weibull distribution with parameters <VAR>alpha</VAR> and <VAR>sigma</VAR>
which must be scalar or of size <VAR>r</VAR> by <VAR>c</VAR>. Or if <VAR>sz</VAR>
is a vector return a matrix of size <VAR>sz</VAR>.
<P>

If <VAR>r</VAR> and <VAR>c</VAR> are omitted, the size of the result matrix is
the common size of <VAR>alpha</VAR> and <VAR>sigma</VAR>.
</P>
</DL>
<P>

<A NAME="doc-wiener_rnd"></A>
<A NAME="IDX944"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>wiener_rnd</B> <I>(<VAR>t</VAR>, <VAR>d</VAR>, <VAR>n</VAR>)</I>
<DD>Return a simulated realization of the <VAR>d</VAR>-dimensional Wiener Process
on the interval [0, <VAR>t</VAR>].  If <VAR>d</VAR> is omitted, <VAR>d</VAR> = 1 is
used. The first column of the return matrix contains time, the
remaining columns contain the Wiener process.
<P>

The optional parameter <VAR>n</VAR> gives the number of summands used for
simulating the process over an interval of length 1.  If <VAR>n</VAR> is
omitted, <VAR>n</VAR> = 1000 is used.
</P>
</DL>
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