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<a name="Normal-Random-Variable"></a>
<div class="header">
<p>
Next: <a href="maxima_222.html#Student_0027s-t-Random-Variable" accesskey="n" rel="next">Student's t Random Variable</a>, Previous: <a href="maxima_220.html#Functions-and-Variables-for-continuous-distributions" accesskey="p" rel="previous">Functions and Variables for continuous distributions</a>, Up: <a href="maxima_220.html#Functions-and-Variables-for-continuous-distributions" accesskey="u" rel="up">Functions and Variables for continuous distributions</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Normal-Random-Variable-1"></a>
<h4 class="subsection">52.2.1 Normal Random Variable</h4>
<p>Normal random variables (also called Gaussian) is denoted
by
\({\it Normal}(m, s)\) where
<em>m</em> is the mean and <em>s > 0</em> is the standard deviation.
</p>
<a name="pdf_005fnormal"></a><a name="Item_003a-distrib_002fdeffn_002fpdf_005fnormal"></a><dl>
<dt><a name="index-pdf_005fnormal"></a>Function: <strong>pdf_normal</strong> <em>(<var>x</var>,<var>m</var>,<var>s</var>)</em></dt>
<dd><p>Returns the value at <var>x</var> of the density function of a
\({\it Normal}(m, s)\) random variable, with <em>s>0</em>. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The pdf is
$$
f(x; m, s) = {1\over s\sqrt{2\pi}} e^{\displaystyle -{(x-m)^2\over 2s^2}}
$$</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="cdf_005fnormal"></a><a name="Item_003a-distrib_002fdeffn_002fcdf_005fnormal"></a><dl>
<dt><a name="index-cdf_005fnormal"></a>Function: <strong>cdf_normal</strong> <em>(<var>x</var>,<var>m</var>,<var>s</var>)</em></dt>
<dd><p>Returns the value at <var>x</var> of the distribution function of a
\({\it Normal}(m, s)\) random variable, with <em>s>0</em>. This function is defined in terms of Maxima’s built-in error function <code>erf</code>.
</p>
<p>The cdf can be written analytically:
$$
F(x; m, s) = {1\over 2} + {1\over 2} {\rm erf}\left(x-m\over s\sqrt{2}\right)
$$</p>
<div class="example">
<pre class="example">(%i1) load ("distrib")$
</pre><pre class="example">(%i2) cdf_normal(x,m,s);
x - m
erf(---------)
sqrt(2) s 1
(%o2) -------------- + -
2 2
</pre></div>
<p>See also <code><a href="maxima_89.html#erf">erf</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
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</dd></dl>
<a name="quantile_005fnormal"></a><a name="Item_003a-distrib_002fdeffn_002fquantile_005fnormal"></a><dl>
<dt><a name="index-quantile_005fnormal"></a>Function: <strong>quantile_normal</strong> <em>(<var>q</var>,<var>m</var>,<var>s</var>)</em></dt>
<dd><p>Returns the <var>q</var>-quantile of a
\({\it Normal}(m, s)\) random variable, with <em>s>0</em>; in other words, this is the inverse of <code><a href="#cdf_005fnormal">cdf_normal</a></code>. Argument <var>q</var> must be an element of <em>[0,1]</em>. To make use of this function, write first <code>load("distrib")</code>.
</p>
<div class="example">
<pre class="example">(%i1) load ("distrib")$
</pre><pre class="example">(%i2) quantile_normal(95/100,0,1);
9
(%o2) sqrt(2) inverse_erf(--)
10
</pre><pre class="example">(%i3) float(%);
(%o3) 1.644853626951472
</pre></div>
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</dd></dl>
<a name="mean_005fnormal"></a><a name="Item_003a-distrib_002fdeffn_002fmean_005fnormal"></a><dl>
<dt><a name="index-mean_005fnormal"></a>Function: <strong>mean_normal</strong> <em>(<var>m</var>,<var>s</var>)</em></dt>
<dd><p>Returns the mean of a
\({\it Normal}(m, s)\) random variable, with <em>s>0</em>. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The mean is
$$
E[X] = m
$$</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="var_005fnormal"></a><a name="Item_003a-distrib_002fdeffn_002fvar_005fnormal"></a><dl>
<dt><a name="index-var_005fnormal"></a>Function: <strong>var_normal</strong> <em>(<var>m</var>,<var>s</var>)</em></dt>
<dd><p>Returns the variance of a
\({\it Normal}(m, s)\) random variable, with <em>s>0</em>. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The variance is
$$
V[X] = s^2
$$</p>
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</dd></dl>
<a name="std_005fnormal"></a><a name="Item_003a-distrib_002fdeffn_002fstd_005fnormal"></a><dl>
<dt><a name="index-std_005fnormal"></a>Function: <strong>std_normal</strong> <em>(<var>m</var>,<var>s</var>)</em></dt>
<dd><p>Returns the standard deviation of a
\({\it Normal}(m, s)\) random variable, with <em>s>0</em>, namely <var>s</var>. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The standard deviation is
$$
D[X] = s
$$</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="skewness_005fnormal"></a><a name="Item_003a-distrib_002fdeffn_002fskewness_005fnormal"></a><dl>
<dt><a name="index-skewness_005fnormal"></a>Function: <strong>skewness_normal</strong> <em>(<var>m</var>,<var>s</var>)</em></dt>
<dd><p>Returns the skewness coefficient of a
\({\it Normal}(m, s)\) random variable, with <em>s>0</em>. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The skewness coefficient is
$$
SK[X] = 0
$$</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="kurtosis_005fnormal"></a><a name="Item_003a-distrib_002fdeffn_002fkurtosis_005fnormal"></a><dl>
<dt><a name="index-kurtosis_005fnormal"></a>Function: <strong>kurtosis_normal</strong> <em>(<var>m</var>,<var>s</var>)</em></dt>
<dd><p>Returns the kurtosis coefficient of a
\({\it Normal}(m, s)\) random variable, with <em>s>0</em>, which is always equal to 0. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The kurtosis coefficient is
$$
KU[X] = 0
$$</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="random_005fnormal"></a><a name="Item_003a-distrib_002fdeffn_002frandom_005fnormal"></a><dl>
<dt><a name="index-random_005fnormal"></a>Function: <strong>random_normal</strong> <em>(<var>m</var>,<var>s</var>) <br> <tt>random_normal</tt> (<var>m</var>,<var>s</var>,<var>n</var>)</em></dt>
<dd>
<p>Returns a
\({\it Normal}(m, s)\) random variate, with <em>s>0</em>. Calling <code>random_normal</code> with a third argument <var>n</var>, a random sample of size <var>n</var> will be simulated.
</p>
<p>This is an implementation of the Box-Mueller algorithm, as described in Knuth, D.E. (1981) <var>Seminumerical Algorithms. The Art of Computer Programming.</var> Addison-Wesley.
</p>
<p>To make use of this function, write first <code>load("distrib")</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·<a href="maxima_424.html#Category_003a-Random-numbers">Random numbers</a>
·</div>
</dd></dl>
<a name="Item_003a-distrib_002fnode_002fStudent_0027s-t-Random-Variable"></a><hr>
<div class="header">
<p>
Next: <a href="maxima_222.html#Student_0027s-t-Random-Variable" accesskey="n" rel="next">Student's t Random Variable</a>, Previous: <a href="maxima_220.html#Functions-and-Variables-for-continuous-distributions" accesskey="p" rel="previous">Functions and Variables for continuous distributions</a>, Up: <a href="maxima_220.html#Functions-and-Variables-for-continuous-distributions" accesskey="u" rel="up">Functions and Variables for continuous distributions</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
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