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<a name="Distributions"></a>
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<h3 class="section">25.6 Distributions</h3>
<p>Octave has functions for computing the Probability Density Function
(PDF), the Cumulative Distribution function (CDF), and the quantile
(the inverse of the CDF) of a large number of distributions.
<p>The following table summarizes the supported distributions (in
alphabetical order).
<!-- Do the table explicitly in TeX if possible to get a better layout. -->
<p><table summary=""><tr align="left"><td valign="top" width="31%"><strong>Distribution</strong>
</td><td valign="top" width="23%"><strong>PDF</strong>
</td><td valign="top" width="23%"><strong>CDF</strong>
</td><td valign="top" width="23%"><strong>Quantile</strong>
<br></td></tr><tr align="left"><td valign="top" width="31%">Beta Distribution
</td><td valign="top" width="23%"><code>betapdf</code>
</td><td valign="top" width="23%"><code>betacdf</code>
</td><td valign="top" width="23%"><code>betainv</code>
<br></td></tr><tr align="left"><td valign="top" width="31%">Binomial Distribution
</td><td valign="top" width="23%"><code>binopdf</code>
</td><td valign="top" width="23%"><code>binocdf</code>
</td><td valign="top" width="23%"><code>binoinv</code>
<br></td></tr><tr align="left"><td valign="top" width="31%">Cauchy Distribution
</td><td valign="top" width="23%"><code>cauchy_pdf</code>
</td><td valign="top" width="23%"><code>cauchy_cdf</code>
</td><td valign="top" width="23%"><code>cauchy_inv</code>
<br></td></tr><tr align="left"><td valign="top" width="31%">Chi-Square Distribution
</td><td valign="top" width="23%"><code>chi2pdf</code>
</td><td valign="top" width="23%"><code>chi2cdf</code>
</td><td valign="top" width="23%"><code>chi2inv</code>
<br></td></tr><tr align="left"><td valign="top" width="31%">Univariate Discrete Distribution
</td><td valign="top" width="23%"><code>discrete_pdf</code>
</td><td valign="top" width="23%"><code>discrete_cdf</code>
</td><td valign="top" width="23%"><code>discrete_inv</code>
<br></td></tr><tr align="left"><td valign="top" width="31%">Empirical Distribution
</td><td valign="top" width="23%"><code>empirical_pdf</code>
</td><td valign="top" width="23%"><code>empirical_cdf</code>
</td><td valign="top" width="23%"><code>empirical_inv</code>
<br></td></tr><tr align="left"><td valign="top" width="31%">Exponential Distribution
</td><td valign="top" width="23%"><code>exppdf</code>
</td><td valign="top" width="23%"><code>expcdf</code>
</td><td valign="top" width="23%"><code>expinv</code>
<br></td></tr><tr align="left"><td valign="top" width="31%">F Distribution
</td><td valign="top" width="23%"><code>fpdf</code>
</td><td valign="top" width="23%"><code>fcdf</code>
</td><td valign="top" width="23%"><code>finv</code>
<br></td></tr><tr align="left"><td valign="top" width="31%">Gamma Distribution
</td><td valign="top" width="23%"><code>gampdf</code>
</td><td valign="top" width="23%"><code>gamcdf</code>
</td><td valign="top" width="23%"><code>gaminv</code>
<br></td></tr><tr align="left"><td valign="top" width="31%">Geometric Distribution
</td><td valign="top" width="23%"><code>geopdf</code>
</td><td valign="top" width="23%"><code>geocdf</code>
</td><td valign="top" width="23%"><code>geoinv</code>
<br></td></tr><tr align="left"><td valign="top" width="31%">Hypergeometric Distribution
</td><td valign="top" width="23%"><code>hygepdf</code>
</td><td valign="top" width="23%"><code>hygecdf</code>
</td><td valign="top" width="23%"><code>hygeinv</code>
<br></td></tr><tr align="left"><td valign="top" width="31%">Kolmogorov Smirnov Distribution
</td><td valign="top" width="23%"><em>Not Available</em>
</td><td valign="top" width="23%"><code>kolmogorov_smirnov_cdf</code>
</td><td valign="top" width="23%"><em>Not Available</em>
<br></td></tr><tr align="left"><td valign="top" width="31%">Laplace Distribution
</td><td valign="top" width="23%"><code>laplace_pdf</code>
</td><td valign="top" width="23%"><code>laplace_cdf</code>
</td><td valign="top" width="23%"><code>laplace_inv</code>
<br></td></tr><tr align="left"><td valign="top" width="31%">Logistic Distribution
</td><td valign="top" width="23%"><code>logistic_pdf</code>
</td><td valign="top" width="23%"><code>logistic_cdf</code>
</td><td valign="top" width="23%"><code>logistic_inv</code>
<br></td></tr><tr align="left"><td valign="top" width="31%">Log-Normal Distribution
</td><td valign="top" width="23%"><code>lognpdf</code>
</td><td valign="top" width="23%"><code>logncdf</code>
</td><td valign="top" width="23%"><code>logninv</code>
<br></td></tr><tr align="left"><td valign="top" width="31%">Pascal Distribution
</td><td valign="top" width="23%"><code>nbinpdf</code>
</td><td valign="top" width="23%"><code>nbincdf</code>
</td><td valign="top" width="23%"><code>nbininv</code>
<br></td></tr><tr align="left"><td valign="top" width="31%">Univariate Normal Distribution
</td><td valign="top" width="23%"><code>normpdf</code>
</td><td valign="top" width="23%"><code>normcdf</code>
</td><td valign="top" width="23%"><code>norminv</code>
<br></td></tr><tr align="left"><td valign="top" width="31%">Poisson Distribution
</td><td valign="top" width="23%"><code>poisspdf</code>
</td><td valign="top" width="23%"><code>poisscdf</code>
</td><td valign="top" width="23%"><code>poissinv</code>
<br></td></tr><tr align="left"><td valign="top" width="31%">t (Student) Distribution
</td><td valign="top" width="23%"><code>tpdf</code>
</td><td valign="top" width="23%"><code>tcdf</code>
</td><td valign="top" width="23%"><code>tinv</code>
<br></td></tr><tr align="left"><td valign="top" width="31%">Univariate Discrete Distribution
</td><td valign="top" width="23%"><code>unidpdf</code>
</td><td valign="top" width="23%"><code>unidcdf</code>
</td><td valign="top" width="23%"><code>unidinv</code>
<br></td></tr><tr align="left"><td valign="top" width="31%">Uniform Distribution
</td><td valign="top" width="23%"><code>unifpdf</code>
</td><td valign="top" width="23%"><code>unifcdf</code>
</td><td valign="top" width="23%"><code>unifinv</code>
<br></td></tr><tr align="left"><td valign="top" width="31%">Weibull Distribution
</td><td valign="top" width="23%"><code>wblpdf</code>
</td><td valign="top" width="23%"><code>wblcdf</code>
</td><td valign="top" width="23%"><code>wblinv</code>
<br></td></tr></table>
<!-- ./statistics/distributions/betacdf.m -->
<p><a name="doc_002dbetacdf"></a>
<div class="defun">
— Function File: <b>betacdf</b> (<var>x, a, b</var>)<var><a name="index-betacdf-1891"></a></var><br>
<blockquote><p>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>) <= <var>x</var>).
</p></blockquote></div>
<!-- ./statistics/distributions/betainv.m -->
<p><a name="doc_002dbetainv"></a>
<div class="defun">
— Function File: <b>betainv</b> (<var>x, a, b</var>)<var><a name="index-betainv-1892"></a></var><br>
<blockquote><p>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>.
</p></blockquote></div>
<!-- ./statistics/distributions/betapdf.m -->
<p><a name="doc_002dbetapdf"></a>
<div class="defun">
— Function File: <b>betapdf</b> (<var>x, a, b</var>)<var><a name="index-betapdf-1893"></a></var><br>
<blockquote><p>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>.
</p></blockquote></div>
<!-- ./statistics/distributions/binocdf.m -->
<p><a name="doc_002dbinocdf"></a>
<div class="defun">
— Function File: <b>binocdf</b> (<var>x, n, p</var>)<var><a name="index-binocdf-1894"></a></var><br>
<blockquote><p>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>.
</p></blockquote></div>
<!-- ./statistics/distributions/binoinv.m -->
<p><a name="doc_002dbinoinv"></a>
<div class="defun">
— Function File: <b>binoinv</b> (<var>x, n, p</var>)<var><a name="index-binoinv-1895"></a></var><br>
<blockquote><p>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>.
</p></blockquote></div>
<!-- ./statistics/distributions/binopdf.m -->
<p><a name="doc_002dbinopdf"></a>
<div class="defun">
— Function File: <b>binopdf</b> (<var>x, n, p</var>)<var><a name="index-binopdf-1896"></a></var><br>
<blockquote><p>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>.
</p></blockquote></div>
<!-- ./statistics/distributions/cauchy_cdf.m -->
<p><a name="doc_002dcauchy_005fcdf"></a>
<div class="defun">
— Function File: <b>cauchy_cdf</b> (<var>x, lambda, sigma</var>)<var><a name="index-cauchy_005fcdf-1897"></a></var><br>
<blockquote><p>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.
</p></blockquote></div>
<!-- ./statistics/distributions/cauchy_inv.m -->
<p><a name="doc_002dcauchy_005finv"></a>
<div class="defun">
— Function File: <b>cauchy_inv</b> (<var>x, lambda, sigma</var>)<var><a name="index-cauchy_005finv-1898"></a></var><br>
<blockquote><p>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.
</p></blockquote></div>
<!-- ./statistics/distributions/cauchy_pdf.m -->
<p><a name="doc_002dcauchy_005fpdf"></a>
<div class="defun">
— Function File: <b>cauchy_pdf</b> (<var>x, lambda, sigma</var>)<var><a name="index-cauchy_005fpdf-1899"></a></var><br>
<blockquote><p>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> > 0. Default values are
<var>lambda</var> = 0, <var>sigma</var> = 1.
</p></blockquote></div>
<!-- ./statistics/distributions/chi2cdf.m -->
<p><a name="doc_002dchi2cdf"></a>
<div class="defun">
— Function File: <b>chi2cdf</b> (<var>x, n</var>)<var><a name="index-chi2cdf-1900"></a></var><br>
<blockquote><p>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.
</p></blockquote></div>
<!-- ./statistics/distributions/chi2inv.m -->
<p><a name="doc_002dchi2inv"></a>
<div class="defun">
— Function File: <b>chi2inv</b> (<var>x, n</var>)<var><a name="index-chi2inv-1901"></a></var><br>
<blockquote><p>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.
</p></blockquote></div>
<!-- ./statistics/distributions/chi2pdf.m -->
<p><a name="doc_002dchi2pdf"></a>
<div class="defun">
— Function File: <b>chisquare_pdf</b> (<var>x, n</var>)<var><a name="index-chisquare_005fpdf-1902"></a></var><br>
<blockquote><p>For each element of <var>x</var>, compute the probability density function
(PDF) at <var>x</var> of the chisquare distribution with <var>n</var> degrees
of freedom.
</p></blockquote></div>
<!-- ./statistics/distributions/discrete_cdf.m -->
<p><a name="doc_002ddiscrete_005fcdf"></a>
<div class="defun">
— Function File: <b>discrete_cdf</b> (<var>x, v, p</var>)<var><a name="index-discrete_005fcdf-1903"></a></var><br>
<blockquote><p>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 <var>v</var> with probabilities <var>p</var>.
</p></blockquote></div>
<!-- ./statistics/distributions/discrete_inv.m -->
<p><a name="doc_002ddiscrete_005finv"></a>
<div class="defun">
— Function File: <b>discrete_inv</b> (<var>x, v, p</var>)<var><a name="index-discrete_005finv-1904"></a></var><br>
<blockquote><p>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>.
</p></blockquote></div>
<!-- ./statistics/distributions/discrete_pdf.m -->
<p><a name="doc_002ddiscrete_005fpdf"></a>
<div class="defun">
— Function File: <b>discrete_pdf</b> (<var>x, v, p</var>)<var><a name="index-discrete_005fpdf-1905"></a></var><br>
<blockquote><p>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>.
</p></blockquote></div>
<!-- ./statistics/distributions/empirical_cdf.m -->
<p><a name="doc_002dempirical_005fcdf"></a>
<div class="defun">
— Function File: <b>empirical_cdf</b> (<var>x, data</var>)<var><a name="index-empirical_005fcdf-1906"></a></var><br>
<blockquote><p>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>.
</p></blockquote></div>
<!-- ./statistics/distributions/empirical_inv.m -->
<p><a name="doc_002dempirical_005finv"></a>
<div class="defun">
— Function File: <b>empirical_inv</b> (<var>x, data</var>)<var><a name="index-empirical_005finv-1907"></a></var><br>
<blockquote><p>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>.
</p></blockquote></div>
<!-- ./statistics/distributions/empirical_pdf.m -->
<p><a name="doc_002dempirical_005fpdf"></a>
<div class="defun">
— Function File: <b>empirical_pdf</b> (<var>x, data</var>)<var><a name="index-empirical_005fpdf-1908"></a></var><br>
<blockquote><p>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>.
</p></blockquote></div>
<!-- ./statistics/distributions/expcdf.m -->
<p><a name="doc_002dexpcdf"></a>
<div class="defun">
— Function File: <b>expcdf</b> (<var>x, lambda</var>)<var><a name="index-expcdf-1909"></a></var><br>
<blockquote><p>For each element of <var>x</var>, compute the cumulative distribution
function (CDF) at <var>x</var> of the exponential distribution with
mean <var>lambda</var>.
<p>The arguments can be of common size or scalar.
</p></blockquote></div>
<!-- ./statistics/distributions/expinv.m -->
<p><a name="doc_002dexpinv"></a>
<div class="defun">
— Function File: <b>expinv</b> (<var>x, lambda</var>)<var><a name="index-expinv-1910"></a></var><br>
<blockquote><p>For each element of <var>x</var>, compute the quantile (the inverse of the
CDF) at <var>x</var> of the exponential distribution with mean
<var>lambda</var>.
</p></blockquote></div>
<!-- ./statistics/distributions/exppdf.m -->
<p><a name="doc_002dexppdf"></a>
<div class="defun">
— Function File: <b>exppdf</b> (<var>x, lambda</var>)<var><a name="index-exppdf-1911"></a></var><br>
<blockquote><p>For each element of <var>x</var>, compute the probability density function
(PDF) of the exponential distribution with mean <var>lambda</var>.
</p></blockquote></div>
<!-- ./statistics/distributions/fcdf.m -->
<p><a name="doc_002dfcdf"></a>
<div class="defun">
— Function File: <b>fcdf</b> (<var>x, m, n</var>)<var><a name="index-fcdf-1912"></a></var><br>
<blockquote><p>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>) <= <var>x</var>).
</p></blockquote></div>
<!-- ./statistics/distributions/finv.m -->
<p><a name="doc_002dfinv"></a>
<div class="defun">
— Function File: <b>finv</b> (<var>x, m, n</var>)<var><a name="index-finv-1913"></a></var><br>
<blockquote><p>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>.
</p></blockquote></div>
<!-- ./statistics/distributions/fpdf.m -->
<p><a name="doc_002dfpdf"></a>
<div class="defun">
— Function File: <b>fpdf</b> (<var>x, m, n</var>)<var><a name="index-fpdf-1914"></a></var><br>
<blockquote><p>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.
</p></blockquote></div>
<!-- ./statistics/distributions/gamcdf.m -->
<p><a name="doc_002dgamcdf"></a>
<div class="defun">
— Function File: <b>gamcdf</b> (<var>x, a, b</var>)<var><a name="index-gamcdf-1915"></a></var><br>
<blockquote><p>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>.
<!-- Texinfo @sp should work but in practice produces ugly results for HTML. -->
<!-- A simple blank line produces the correct behavior. -->
<!-- @sp 1 -->
<p class="noindent"><strong>See also:</strong> <a href="doc_002dgamma.html#doc_002dgamma">gamma</a>, <a href="doc_002dgammaln.html#doc_002dgammaln">gammaln</a>, <a href="doc_002dgammainc.html#doc_002dgammainc">gammainc</a>, <a href="doc_002dgampdf.html#doc_002dgampdf">gampdf</a>, <a href="doc_002dgaminv.html#doc_002dgaminv">gaminv</a>, <a href="doc_002dgamrnd.html#doc_002dgamrnd">gamrnd</a>.
</p></blockquote></div>
<!-- ./statistics/distributions/gaminv.m -->
<p><a name="doc_002dgaminv"></a>
<div class="defun">
— Function File: <b>gaminv</b> (<var>x, a, b</var>)<var><a name="index-gaminv-1916"></a></var><br>
<blockquote><p>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>.
<!-- Texinfo @sp should work but in practice produces ugly results for HTML. -->
<!-- A simple blank line produces the correct behavior. -->
<!-- @sp 1 -->
<p class="noindent"><strong>See also:</strong> <a href="doc_002dgamma.html#doc_002dgamma">gamma</a>, <a href="doc_002dgammaln.html#doc_002dgammaln">gammaln</a>, <a href="doc_002dgammainc.html#doc_002dgammainc">gammainc</a>, <a href="doc_002dgampdf.html#doc_002dgampdf">gampdf</a>, <a href="doc_002dgamcdf.html#doc_002dgamcdf">gamcdf</a>, <a href="doc_002dgamrnd.html#doc_002dgamrnd">gamrnd</a>.
</p></blockquote></div>
<!-- ./statistics/distributions/gampdf.m -->
<p><a name="doc_002dgampdf"></a>
<div class="defun">
— Function File: <b>gampdf</b> (<var>x, a, b</var>)<var><a name="index-gampdf-1917"></a></var><br>
<blockquote><p>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>.
<!-- Texinfo @sp should work but in practice produces ugly results for HTML. -->
<!-- A simple blank line produces the correct behavior. -->
<!-- @sp 1 -->
<p class="noindent"><strong>See also:</strong> <a href="doc_002dgamma.html#doc_002dgamma">gamma</a>, <a href="doc_002dgammaln.html#doc_002dgammaln">gammaln</a>, <a href="doc_002dgammainc.html#doc_002dgammainc">gammainc</a>, <a href="doc_002dgamcdf.html#doc_002dgamcdf">gamcdf</a>, <a href="doc_002dgaminv.html#doc_002dgaminv">gaminv</a>, <a href="doc_002dgamrnd.html#doc_002dgamrnd">gamrnd</a>.
</p></blockquote></div>
<!-- ./statistics/distributions/geocdf.m -->
<p><a name="doc_002dgeocdf"></a>
<div class="defun">
— Function File: <b>geocdf</b> (<var>x, p</var>)<var><a name="index-geocdf-1918"></a></var><br>
<blockquote><p>For each element of <var>x</var>, compute the CDF at <var>x</var> of the
geometric distribution with parameter <var>p</var>.
</p></blockquote></div>
<!-- ./statistics/distributions/geoinv.m -->
<p><a name="doc_002dgeoinv"></a>
<div class="defun">
— Function File: <b>geoinv</b> (<var>x, p</var>)<var><a name="index-geoinv-1919"></a></var><br>
<blockquote><p>For each element of <var>x</var>, compute the quantile at <var>x</var> of the
geometric distribution with parameter <var>p</var>.
</p></blockquote></div>
<!-- ./statistics/distributions/geopdf.m -->
<p><a name="doc_002dgeopdf"></a>
<div class="defun">
— Function File: <b>geopdf</b> (<var>x, p</var>)<var><a name="index-geopdf-1920"></a></var><br>
<blockquote><p>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>.
</p></blockquote></div>
<!-- ./statistics/distributions/hygecdf.m -->
<p><a name="doc_002dhygecdf"></a>
<div class="defun">
— Function File: <b>hygecdf</b> (<var>x, t, m, n</var>)<var><a name="index-hygecdf-1921"></a></var><br>
<blockquote><p>Compute the cumulative distribution function (CDF) at <var>x</var> of the
hypergeometric distribution with parameters <var>t</var>, <var>m</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>t</var>, <var>m</var>, and <var>n</var> must positive integers
with <var>m</var> and <var>n</var> not greater than <var>t</var>.
</p></blockquote></div>
<!-- ./statistics/distributions/hygeinv.m -->
<p><a name="doc_002dhygeinv"></a>
<div class="defun">
— Function File: <b>hygeinv</b> (<var>x, t, m, n</var>)<var><a name="index-hygeinv-1922"></a></var><br>
<blockquote><p>For each element of <var>x</var>, compute the quantile at <var>x</var> of the
hypergeometric distribution with parameters <var>t</var>, <var>m</var>, and
<var>n</var>.
<p>The parameters <var>t</var>, <var>m</var>, and <var>n</var> must positive integers
with <var>m</var> and <var>n</var> not greater than <var>t</var>.
</p></blockquote></div>
<!-- ./statistics/distributions/hygepdf.m -->
<p><a name="doc_002dhygepdf"></a>
<div class="defun">
— Function File: <b>hygepdf</b> (<var>x, t, m, n</var>)<var><a name="index-hygepdf-1923"></a></var><br>
<blockquote><p>Compute the probability density function (PDF) at <var>x</var> of the
hypergeometric distribution with parameters <var>t</var>, <var>m</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></blockquote></div>
<!-- ./statistics/distributions/kolmogorov_smirnov_cdf.m -->
<p><a name="doc_002dkolmogorov_005fsmirnov_005fcdf"></a>
<div class="defun">
— Function File: <b>kolmogorov_smirnov_cdf</b> (<var>x, tol</var>)<var><a name="index-kolmogorov_005fsmirnov_005fcdf-1924"></a></var><br>
<blockquote><p>Return the CDF at <var>x</var> of the Kolmogorov-Smirnov distribution,
<pre class="example"> Inf
Q(x) = SUM (-1)^k exp(-2 k^2 x^2)
k = -Inf
</pre>
<p class="noindent">for <var>x</var> > 0.
<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></blockquote></div>
<!-- ./statistics/distributions/laplace_cdf.m -->
<p><a name="doc_002dlaplace_005fcdf"></a>
<div class="defun">
— Function File: <b>laplace_cdf</b> (<var>x</var>)<var><a name="index-laplace_005fcdf-1925"></a></var><br>
<blockquote><p>For each element of <var>x</var>, compute the cumulative distribution
function (CDF) at <var>x</var> of the Laplace distribution.
</p></blockquote></div>
<!-- ./statistics/distributions/laplace_inv.m -->
<p><a name="doc_002dlaplace_005finv"></a>
<div class="defun">
— Function File: <b>laplace_inv</b> (<var>x</var>)<var><a name="index-laplace_005finv-1926"></a></var><br>
<blockquote><p>For each element of <var>x</var>, compute the quantile (the inverse of the
CDF) at <var>x</var> of the Laplace distribution.
</p></blockquote></div>
<!-- ./statistics/distributions/laplace_pdf.m -->
<p><a name="doc_002dlaplace_005fpdf"></a>
<div class="defun">
— Function File: <b>laplace_pdf</b> (<var>x</var>)<var><a name="index-laplace_005fpdf-1927"></a></var><br>
<blockquote><p>For each element of <var>x</var>, compute the probability density function
(PDF) at <var>x</var> of the Laplace distribution.
</p></blockquote></div>
<!-- ./statistics/distributions/logistic_cdf.m -->
<p><a name="doc_002dlogistic_005fcdf"></a>
<div class="defun">
— Function File: <b>logistic_cdf</b> (<var>x</var>)<var><a name="index-logistic_005fcdf-1928"></a></var><br>
<blockquote><p>For each component of <var>x</var>, compute the CDF at <var>x</var> of the
logistic distribution.
</p></blockquote></div>
<!-- ./statistics/distributions/logistic_inv.m -->
<p><a name="doc_002dlogistic_005finv"></a>
<div class="defun">
— Function File: <b>logistic_inv</b> (<var>x</var>)<var><a name="index-logistic_005finv-1929"></a></var><br>
<blockquote><p>For each component of <var>x</var>, compute the quantile (the inverse of
the CDF) at <var>x</var> of the logistic distribution.
</p></blockquote></div>
<!-- ./statistics/distributions/logistic_pdf.m -->
<p><a name="doc_002dlogistic_005fpdf"></a>
<div class="defun">
— Function File: <b>logistic_pdf</b> (<var>x</var>)<var><a name="index-logistic_005fpdf-1930"></a></var><br>
<blockquote><p>For each component of <var>x</var>, compute the PDF at <var>x</var> of the
logistic distribution.
</p></blockquote></div>
<!-- ./statistics/distributions/logncdf.m -->
<p><a name="doc_002dlogncdf"></a>
<div class="defun">
— Function File: <b>logncdf</b> (<var>x, mu, sigma</var>)<var><a name="index-logncdf-1931"></a></var><br>
<blockquote><p>For each element of <var>x</var>, compute the cumulative distribution
function (CDF) at <var>x</var> of the lognormal distribution with
parameters <var>mu</var> and <var>sigma</var>. If a random variable follows this
distribution, its logarithm is normally distributed with mean
<var>mu</var> and standard deviation <var>sigma</var>.
<p>Default values are <var>mu</var> = 1, <var>sigma</var> = 1.
</p></blockquote></div>
<!-- ./statistics/distributions/logninv.m -->
<p><a name="doc_002dlogninv"></a>
<div class="defun">
— Function File: <b>logninv</b> (<var>x, mu, sigma</var>)<var><a name="index-logninv-1932"></a></var><br>
<blockquote><p>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>mu</var>
and <var>sigma</var>. If a random variable follows this distribution, its
logarithm is normally distributed with mean <code>log (</code><var>mu</var><code>)</code> and
variance <var>sigma</var>.
<p>Default values are <var>mu</var> = 1, <var>sigma</var> = 1.
</p></blockquote></div>
<!-- ./statistics/distributions/lognpdf.m -->
<p><a name="doc_002dlognpdf"></a>
<div class="defun">
— Function File: <b>lognpdf</b> (<var>x, mu, sigma</var>)<var><a name="index-lognpdf-1933"></a></var><br>
<blockquote><p>For each element of <var>x</var>, compute the probability density function
(PDF) at <var>x</var> of the lognormal distribution with parameters
<var>mu</var> and <var>sigma</var>. If a random variable follows this distribution,
its logarithm is normally distributed with mean <var>mu</var>
and standard deviation <var>sigma</var>.
<p>Default values are <var>mu</var> = 1, <var>sigma</var> = 1.
</p></blockquote></div>
<!-- ./statistics/distributions/nbincdf.m -->
<p><a name="doc_002dnbincdf"></a>
<div class="defun">
— Function File: <b>nbincdf</b> (<var>x, n, p</var>)<var><a name="index-nbincdf-1934"></a></var><br>
<blockquote><p>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></blockquote></div>
<!-- ./statistics/distributions/nbininv.m -->
<p><a name="doc_002dnbininv"></a>
<div class="defun">
— Function File: <b>nbininv</b> (<var>x, n, p</var>)<var><a name="index-nbininv-1935"></a></var><br>
<blockquote><p>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></blockquote></div>
<!-- ./statistics/distributions/nbinpdf.m -->
<p><a name="doc_002dnbinpdf"></a>
<div class="defun">
— Function File: <b>nbinpdf</b> (<var>x, n, p</var>)<var><a name="index-nbinpdf-1936"></a></var><br>
<blockquote><p>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></blockquote></div>
<!-- ./statistics/distributions/normcdf.m -->
<p><a name="doc_002dnormcdf"></a>
<div class="defun">
— Function File: <b>normcdf</b> (<var>x, m, s</var>)<var><a name="index-normcdf-1937"></a></var><br>
<blockquote><p>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 standard deviation <var>s</var>.
<p>Default values are <var>m</var> = 0, <var>s</var> = 1.
</p></blockquote></div>
<!-- ./statistics/distributions/norminv.m -->
<p><a name="doc_002dnorminv"></a>
<div class="defun">
— Function File: <b>norminv</b> (<var>x, m, s</var>)<var><a name="index-norminv-1938"></a></var><br>
<blockquote><p>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
standard deviation <var>s</var>.
<p>Default values are <var>m</var> = 0, <var>s</var> = 1.
</p></blockquote></div>
<!-- ./statistics/distributions/normpdf.m -->
<p><a name="doc_002dnormpdf"></a>
<div class="defun">
— Function File: <b>normpdf</b> (<var>x, m, s</var>)<var><a name="index-normpdf-1939"></a></var><br>
<blockquote><p>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
standard deviation <var>s</var>.
<p>Default values are <var>m</var> = 0, <var>s</var> = 1.
</p></blockquote></div>
<!-- ./statistics/distributions/poisscdf.m -->
<p><a name="doc_002dpoisscdf"></a>
<div class="defun">
— Function File: <b>poisscdf</b> (<var>x, lambda</var>)<var><a name="index-poisscdf-1940"></a></var><br>
<blockquote><p>For each element of <var>x</var>, compute the cumulative distribution
function (CDF) at <var>x</var> of the Poisson distribution with parameter
lambda.
</p></blockquote></div>
<!-- ./statistics/distributions/poissinv.m -->
<p><a name="doc_002dpoissinv"></a>
<div class="defun">
— Function File: <b>poissinv</b> (<var>x, lambda</var>)<var><a name="index-poissinv-1941"></a></var><br>
<blockquote><p>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>.
</p></blockquote></div>
<!-- ./statistics/distributions/poisspdf.m -->
<p><a name="doc_002dpoisspdf"></a>
<div class="defun">
— Function File: <b>poisspdf</b> (<var>x, lambda</var>)<var><a name="index-poisspdf-1942"></a></var><br>
<blockquote><p>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>.
</p></blockquote></div>
<!-- ./statistics/distributions/tcdf.m -->
<p><a name="doc_002dtcdf"></a>
<div class="defun">
— Function File: <b>tcdf</b> (<var>x, n</var>)<var><a name="index-tcdf-1943"></a></var><br>
<blockquote><p>For each element of <var>x</var>, compute the cumulative distribution
function (CDF) at <var>x</var> of the t (Student) distribution with
<var>n</var> degrees of freedom, i.e., PROB (t(<var>n</var>) <= <var>x</var>).
</p></blockquote></div>
<!-- ./statistics/distributions/tinv.m -->
<p><a name="doc_002dtinv"></a>
<div class="defun">
— Function File: <b>tinv</b> (<var>x, n</var>)<var><a name="index-tinv-1944"></a></var><br>
<blockquote><p>For each probability value <var>x</var>, compute the inverse of the
cumulative distribution function (CDF) of the t (Student)
distribution with degrees of freedom <var>n</var>. This function is
analogous to looking in a table for the t-value of a single-tailed
distribution.
</p></blockquote></div>
<!-- ./statistics/distributions/tpdf.m -->
<p><a name="doc_002dtpdf"></a>
<div class="defun">
— Function File: <b>tpdf</b> (<var>x, n</var>)<var><a name="index-tpdf-1945"></a></var><br>
<blockquote><p>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.
</p></blockquote></div>
<!-- ./statistics/distributions/unidcdf.m -->
<p><a name="doc_002dunidcdf"></a>
<div class="defun">
— Function File: <b>unidcdf</b> (<var>x, v</var>)<var><a name="index-unidcdf-1946"></a></var><br>
<blockquote><p>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 <var>v</var> with equal probability.
</p></blockquote></div>
<!-- ./statistics/distributions/unidinv.m -->
<p><a name="doc_002dunidinv"></a>
<div class="defun">
— Function File: <b>unidinv</b> (<var>x, v</var>)<var><a name="index-unidinv-1947"></a></var><br>
<blockquote><p>For each component of <var>x</var>, compute the quantile (the inverse of
the CDF) at <var>x</var> of the univariate discrete distribution which assumes the
values in <var>v</var> with equal probability
</p></blockquote></div>
<!-- ./statistics/distributions/unidpdf.m -->
<p><a name="doc_002dunidpdf"></a>
<div class="defun">
— Function File: <b>unidpdf</b> (<var>x, v</var>)<var><a name="index-unidpdf-1948"></a></var><br>
<blockquote><p>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 equal probability.
</p></blockquote></div>
<!-- ./statistics/distributions/unifcdf.m -->
<p><a name="doc_002dunifcdf"></a>
<div class="defun">
— Function File: <b>unifcdf</b> (<var>x, a, b</var>)<var><a name="index-unifcdf-1949"></a></var><br>
<blockquote><p>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>) <= x).
<p>Default values are <var>a</var> = 0, <var>b</var> = 1.
</p></blockquote></div>
<!-- ./statistics/distributions/unifinv.m -->
<p><a name="doc_002dunifinv"></a>
<div class="defun">
— Function File: <b>unifinv</b> (<var>x, a, b</var>)<var><a name="index-unifinv-1950"></a></var><br>
<blockquote><p>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></blockquote></div>
<!-- ./statistics/distributions/unifpdf.m -->
<p><a name="doc_002dunifpdf"></a>
<div class="defun">
— Function File: <b>unifpdf</b> (<var>x, a, b</var>)<var><a name="index-unifpdf-1951"></a></var><br>
<blockquote><p>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></blockquote></div>
<!-- ./statistics/distributions/wblcdf.m -->
<p><a name="doc_002dwblcdf"></a>
<div class="defun">
— Function File: <b>wblcdf</b> (<var>x, scale, shape</var>)<var><a name="index-wblcdf-1952"></a></var><br>
<blockquote><p>Compute the cumulative distribution function (CDF) at <var>x</var> of the
Weibull distribution with shape parameter <var>scale</var> and scale
parameter <var>shape</var>, which is
<pre class="example"> 1 - exp(-(x/shape)^scale)
</pre>
<p>for <var>x</var> >= 0.
</p></blockquote></div>
<!-- ./statistics/distributions/wblinv.m -->
<p><a name="doc_002dwblinv"></a>
<div class="defun">
— Function File: <b>wblinv</b> (<var>x, scale, shape</var>)<var><a name="index-wblinv-1953"></a></var><br>
<blockquote><p>Compute the quantile (the inverse of the CDF) at <var>x</var> of the
Weibull distribution with shape parameter <var>scale</var> and scale
parameter <var>shape</var>.
</p></blockquote></div>
<!-- ./statistics/distributions/wblpdf.m -->
<p><a name="doc_002dwblpdf"></a>
<div class="defun">
— Function File: <b>wblpdf</b> (<var>x, scale, shape</var>)<var><a name="index-wblpdf-1954"></a></var><br>
<blockquote><p>Compute the probability density function (PDF) at <var>x</var> of the
Weibull distribution with shape parameter <var>scale</var> and scale
parameter <var>shape</var> which is given by
<pre class="example"> scale * shape^(-scale) * x^(scale-1) * exp(-(x/shape)^scale)
</pre>
<p class="noindent">for <var>x</var> > 0.
</p></blockquote></div>
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