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<a name="Distributions"></a>
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<hr>
<a name="Distributions-1"></a>
<h3 class="section">26.5 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) for a large number of distributions.
</p>
<p>The following table summarizes the supported distributions (in
alphabetical order).
</p>
<table>
<thead><tr><th width="31%">Distribution</th><th width="23%">PDF</th><th width="23%">CDF</th><th width="23%">Quantile</th></tr></thead>
<tr><td width="31%">Beta Distribution</td><td width="23%"><code>betapdf</code></td><td width="23%"><code>betacdf</code></td><td width="23%"><code>betainv</code></td></tr>
<tr><td width="31%">Binomial Distribution</td><td width="23%"><code>binopdf</code></td><td width="23%"><code>binocdf</code></td><td width="23%"><code>binoinv</code></td></tr>
<tr><td width="31%">Cauchy Distribution</td><td width="23%"><code>cauchy_pdf</code></td><td width="23%"><code>cauchy_cdf</code></td><td width="23%"><code>cauchy_inv</code></td></tr>
<tr><td width="31%">Chi-Square Distribution</td><td width="23%"><code>chi2pdf</code></td><td width="23%"><code>chi2cdf</code></td><td width="23%"><code>chi2inv</code></td></tr>
<tr><td width="31%">Univariate Discrete Distribution</td><td width="23%"><code>discrete_pdf</code></td><td width="23%"><code>discrete_cdf</code></td><td width="23%"><code>discrete_inv</code></td></tr>
<tr><td width="31%">Empirical Distribution</td><td width="23%"><code>empirical_pdf</code></td><td width="23%"><code>empirical_cdf</code></td><td width="23%"><code>empirical_inv</code></td></tr>
<tr><td width="31%">Exponential Distribution</td><td width="23%"><code>exppdf</code></td><td width="23%"><code>expcdf</code></td><td width="23%"><code>expinv</code></td></tr>
<tr><td width="31%">F Distribution</td><td width="23%"><code>fpdf</code></td><td width="23%"><code>fcdf</code></td><td width="23%"><code>finv</code></td></tr>
<tr><td width="31%">Gamma Distribution</td><td width="23%"><code>gampdf</code></td><td width="23%"><code>gamcdf</code></td><td width="23%"><code>gaminv</code></td></tr>
<tr><td width="31%">Geometric Distribution</td><td width="23%"><code>geopdf</code></td><td width="23%"><code>geocdf</code></td><td width="23%"><code>geoinv</code></td></tr>
<tr><td width="31%">Hypergeometric Distribution</td><td width="23%"><code>hygepdf</code></td><td width="23%"><code>hygecdf</code></td><td width="23%"><code>hygeinv</code></td></tr>
<tr><td width="31%">Kolmogorov Smirnov Distribution</td><td width="23%"><em>Not Available</em></td><td width="23%"><code>kolmogorov_smirnov_cdf</code></td><td width="23%"><em>Not Available</em></td></tr>
<tr><td width="31%">Laplace Distribution</td><td width="23%"><code>laplace_pdf</code></td><td width="23%"><code>laplace_cdf</code></td><td width="23%"><code>laplace_inv</code></td></tr>
<tr><td width="31%">Logistic Distribution</td><td width="23%"><code>logistic_pdf</code></td><td width="23%"><code>logistic_cdf</code></td><td width="23%"><code>logistic_inv</code></td></tr>
<tr><td width="31%">Log-Normal Distribution</td><td width="23%"><code>lognpdf</code></td><td width="23%"><code>logncdf</code></td><td width="23%"><code>logninv</code></td></tr>
<tr><td width="31%">Univariate Normal Distribution</td><td width="23%"><code>normpdf</code></td><td width="23%"><code>normcdf</code></td><td width="23%"><code>norminv</code></td></tr>
<tr><td width="31%">Pascal Distribution</td><td width="23%"><code>nbinpdf</code></td><td width="23%"><code>nbincdf</code></td><td width="23%"><code>nbininv</code></td></tr>
<tr><td width="31%">Poisson Distribution</td><td width="23%"><code>poisspdf</code></td><td width="23%"><code>poisscdf</code></td><td width="23%"><code>poissinv</code></td></tr>
<tr><td width="31%">Standard Normal Distribution</td><td width="23%"><code>stdnormal_pdf</code></td><td width="23%"><code>stdnormal_cdf</code></td><td width="23%"><code>stdnormal_inv</code></td></tr>
<tr><td width="31%">t (Student) Distribution</td><td width="23%"><code>tpdf</code></td><td width="23%"><code>tcdf</code></td><td width="23%"><code>tinv</code></td></tr>
<tr><td width="31%">Univariate Discrete Distribution</td><td width="23%"><code>unidpdf</code></td><td width="23%"><code>unidcdf</code></td><td width="23%"><code>unidinv</code></td></tr>
<tr><td width="31%">Uniform Distribution</td><td width="23%"><code>unifpdf</code></td><td width="23%"><code>unifcdf</code></td><td width="23%"><code>unifinv</code></td></tr>
<tr><td width="31%">Weibull Distribution</td><td width="23%"><code>wblpdf</code></td><td width="23%"><code>wblcdf</code></td><td width="23%"><code>wblinv</code></td></tr>
</table>

<a name="XREFbetapdf"></a><dl>
<dt><a name="index-betapdf"></a>: <em></em> <strong>betapdf</strong> <em>(<var>x</var>, <var>a</var>, <var>b</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the probability density function (PDF)
at <var>x</var> of the Beta distribution with parameters <var>a</var> and <var>b</var>.
</p></dd></dl>


<a name="XREFbetacdf"></a><dl>
<dt><a name="index-betacdf"></a>: <em></em> <strong>betacdf</strong> <em>(<var>x</var>, <var>a</var>, <var>b</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the cumulative distribution function
(CDF) at <var>x</var> of the Beta distribution with parameters <var>a</var> and
<var>b</var>.
</p></dd></dl>


<a name="XREFbetainv"></a><dl>
<dt><a name="index-betainv"></a>: <em></em> <strong>betainv</strong> <em>(<var>x</var>, <var>a</var>, <var>b</var>)</em></dt>
<dd><p>For each element 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></dd></dl>


<a name="XREFbinopdf"></a><dl>
<dt><a name="index-binopdf"></a>: <em></em> <strong>binopdf</strong> <em>(<var>x</var>, <var>n</var>, <var>p</var>)</em></dt>
<dd><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>,
where <var>n</var> is the number of trials and <var>p</var> is the probability of
success.
</p></dd></dl>


<a name="XREFbinocdf"></a><dl>
<dt><a name="index-binocdf"></a>: <em></em> <strong>binocdf</strong> <em>(<var>x</var>, <var>n</var>, <var>p</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the cumulative distribution function
(CDF) at <var>x</var> of the binomial distribution with parameters <var>n</var> and
<var>p</var>, where <var>n</var> is the number of trials and <var>p</var> is the
probability of success.
</p></dd></dl>


<a name="XREFbinoinv"></a><dl>
<dt><a name="index-binoinv"></a>: <em></em> <strong>binoinv</strong> <em>(<var>x</var>, <var>n</var>, <var>p</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the quantile (the inverse of the CDF)
at <var>x</var> of the binomial distribution with parameters
<var>n</var> and <var>p</var>, where <var>n</var> is the number of trials and
<var>p</var> is the probability of success.
</p></dd></dl>


<a name="XREFcauchy_005fpdf"></a><dl>
<dt><a name="index-cauchy_005fpdf"></a>: <em></em> <strong>cauchy_pdf</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-cauchy_005fpdf-1"></a>: <em></em> <strong>cauchy_pdf</strong> <em>(<var>x</var>, <var>location</var>, <var>scale</var>)</em></dt>
<dd><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>location</var> and scale parameter <var>scale</var> &gt; 0.
</p>
<p>Default values are <var>location</var> = 0, <var>scale</var> = 1.
</p></dd></dl>


<a name="XREFcauchy_005fcdf"></a><dl>
<dt><a name="index-cauchy_005fcdf"></a>: <em></em> <strong>cauchy_cdf</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-cauchy_005fcdf-1"></a>: <em></em> <strong>cauchy_cdf</strong> <em>(<var>x</var>, <var>location</var>, <var>scale</var>)</em></dt>
<dd><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>location</var> and scale parameter <var>scale</var>.
</p>
<p>Default values are <var>location</var> = 0, <var>scale</var> = 1.
</p></dd></dl>


<a name="XREFcauchy_005finv"></a><dl>
<dt><a name="index-cauchy_005finv"></a>: <em></em> <strong>cauchy_inv</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-cauchy_005finv-1"></a>: <em></em> <strong>cauchy_inv</strong> <em>(<var>x</var>, <var>location</var>, <var>scale</var>)</em></dt>
<dd><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>location</var> and scale parameter <var>scale</var>.
</p>
<p>Default values are <var>location</var> = 0, <var>scale</var> = 1.
</p></dd></dl>


<a name="XREFchi2pdf"></a><dl>
<dt><a name="index-chi2pdf"></a>: <em></em> <strong>chi2pdf</strong> <em>(<var>x</var>, <var>n</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the probability density function (PDF)
at <var>x</var> of the chi-square distribution with <var>n</var> degrees of freedom.
</p></dd></dl>


<a name="XREFchi2cdf"></a><dl>
<dt><a name="index-chi2cdf"></a>: <em></em> <strong>chi2cdf</strong> <em>(<var>x</var>, <var>n</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the cumulative distribution function
(CDF) at <var>x</var> of the chi-square distribution with <var>n</var> degrees of
freedom.
</p></dd></dl>


<a name="XREFchi2inv"></a><dl>
<dt><a name="index-chi2inv"></a>: <em></em> <strong>chi2inv</strong> <em>(<var>x</var>, <var>n</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the quantile (the inverse of the CDF)
at <var>x</var> of the chi-square distribution with <var>n</var> degrees of freedom.
</p></dd></dl>


<a name="XREFdiscrete_005fpdf"></a><dl>
<dt><a name="index-discrete_005fpdf"></a>: <em></em> <strong>discrete_pdf</strong> <em>(<var>x</var>, <var>v</var>, <var>p</var>)</em></dt>
<dd><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></dd></dl>


<a name="XREFdiscrete_005fcdf"></a><dl>
<dt><a name="index-discrete_005fcdf"></a>: <em></em> <strong>discrete_cdf</strong> <em>(<var>x</var>, <var>v</var>, <var>p</var>)</em></dt>
<dd><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></dd></dl>


<a name="XREFdiscrete_005finv"></a><dl>
<dt><a name="index-discrete_005finv"></a>: <em></em> <strong>discrete_inv</strong> <em>(<var>x</var>, <var>v</var>, <var>p</var>)</em></dt>
<dd><p>For each element 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></dd></dl>


<a name="XREFempirical_005fpdf"></a><dl>
<dt><a name="index-empirical_005fpdf"></a>: <em></em> <strong>empirical_pdf</strong> <em>(<var>x</var>, <var>data</var>)</em></dt>
<dd><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></dd></dl>


<a name="XREFempirical_005fcdf"></a><dl>
<dt><a name="index-empirical_005fcdf"></a>: <em></em> <strong>empirical_cdf</strong> <em>(<var>x</var>, <var>data</var>)</em></dt>
<dd><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></dd></dl>


<a name="XREFempirical_005finv"></a><dl>
<dt><a name="index-empirical_005finv"></a>: <em></em> <strong>empirical_inv</strong> <em>(<var>x</var>, <var>data</var>)</em></dt>
<dd><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></dd></dl>


<a name="XREFexppdf"></a><dl>
<dt><a name="index-exppdf"></a>: <em></em> <strong>exppdf</strong> <em>(<var>x</var>, <var>lambda</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the probability density function (PDF)
at <var>x</var> of the exponential distribution with mean <var>lambda</var>.
</p></dd></dl>


<a name="XREFexpcdf"></a><dl>
<dt><a name="index-expcdf"></a>: <em></em> <strong>expcdf</strong> <em>(<var>x</var>, <var>lambda</var>)</em></dt>
<dd><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>
<p>The arguments can be of common size or scalars.
</p></dd></dl>


<a name="XREFexpinv"></a><dl>
<dt><a name="index-expinv"></a>: <em></em> <strong>expinv</strong> <em>(<var>x</var>, <var>lambda</var>)</em></dt>
<dd><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></dd></dl>


<a name="XREFfpdf"></a><dl>
<dt><a name="index-fpdf"></a>: <em></em> <strong>fpdf</strong> <em>(<var>x</var>, <var>m</var>, <var>n</var>)</em></dt>
<dd><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></dd></dl>


<a name="XREFfcdf"></a><dl>
<dt><a name="index-fcdf"></a>: <em></em> <strong>fcdf</strong> <em>(<var>x</var>, <var>m</var>, <var>n</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the cumulative distribution function
(CDF) at <var>x</var> of the F distribution with <var>m</var> and <var>n</var> degrees of
freedom.
</p></dd></dl>


<a name="XREFfinv"></a><dl>
<dt><a name="index-finv"></a>: <em></em> <strong>finv</strong> <em>(<var>x</var>, <var>m</var>, <var>n</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the quantile (the inverse of the CDF)
at <var>x</var> of the F distribution with <var>m</var> and <var>n</var> degrees of
freedom.
</p></dd></dl>


<a name="XREFgampdf"></a><dl>
<dt><a name="index-gampdf"></a>: <em></em> <strong>gampdf</strong> <em>(<var>x</var>, <var>a</var>, <var>b</var>)</em></dt>
<dd><p>For each element of <var>x</var>, return the probability density function
(PDF) at <var>x</var> of the Gamma distribution with shape parameter <var>a</var> and
scale <var>b</var>.
</p></dd></dl>


<a name="XREFgamcdf"></a><dl>
<dt><a name="index-gamcdf"></a>: <em></em> <strong>gamcdf</strong> <em>(<var>x</var>, <var>a</var>, <var>b</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the cumulative distribution function
(CDF) at <var>x</var> of the Gamma distribution with shape parameter <var>a</var> and
scale <var>b</var>.
</p></dd></dl>


<a name="XREFgaminv"></a><dl>
<dt><a name="index-gaminv"></a>: <em></em> <strong>gaminv</strong> <em>(<var>x</var>, <var>a</var>, <var>b</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the quantile (the inverse of the CDF)
at <var>x</var> of the Gamma distribution with shape parameter <var>a</var> and
scale <var>b</var>.
</p></dd></dl>


<a name="XREFgeopdf"></a><dl>
<dt><a name="index-geopdf"></a>: <em></em> <strong>geopdf</strong> <em>(<var>x</var>, <var>p</var>)</em></dt>
<dd><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>
<p>The geometric distribution models the number of failures (<var>x</var>-1) of a
Bernoulli trial with probability <var>p</var> before the first success (<var>x</var>).
</p></dd></dl>


<a name="XREFgeocdf"></a><dl>
<dt><a name="index-geocdf"></a>: <em></em> <strong>geocdf</strong> <em>(<var>x</var>, <var>p</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the cumulative distribution function
(CDF) at <var>x</var> of the geometric distribution with parameter <var>p</var>.
</p>
<p>The geometric distribution models the number of failures (<var>x</var>-1) of a
Bernoulli trial with probability <var>p</var> before the first success (<var>x</var>).
</p></dd></dl>


<a name="XREFgeoinv"></a><dl>
<dt><a name="index-geoinv"></a>: <em></em> <strong>geoinv</strong> <em>(<var>x</var>, <var>p</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the quantile (the inverse of the CDF)
at <var>x</var> of the geometric distribution with parameter <var>p</var>.
</p>
<p>The geometric distribution models the number of failures (<var>x</var>-1) of a
Bernoulli trial with probability <var>p</var> before the first success (<var>x</var>).
</p></dd></dl>


<a name="XREFhygepdf"></a><dl>
<dt><a name="index-hygepdf"></a>: <em></em> <strong>hygepdf</strong> <em>(<var>x</var>, <var>t</var>, <var>m</var>, <var>n</var>)</em></dt>
<dd><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>.
</p>
<p>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>
<p>The parameters <var>t</var>, <var>m</var>, and <var>n</var> must be positive integers
with <var>m</var> and <var>n</var> not greater than <var>t</var>.
</p></dd></dl>


<a name="XREFhygecdf"></a><dl>
<dt><a name="index-hygecdf"></a>: <em></em> <strong>hygecdf</strong> <em>(<var>x</var>, <var>t</var>, <var>m</var>, <var>n</var>)</em></dt>
<dd><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>.
</p>
<p>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>
<p>The parameters <var>t</var>, <var>m</var>, and <var>n</var> must be positive integers
with <var>m</var> and <var>n</var> not greater than <var>t</var>.
</p></dd></dl>


<a name="XREFhygeinv"></a><dl>
<dt><a name="index-hygeinv"></a>: <em></em> <strong>hygeinv</strong> <em>(<var>x</var>, <var>t</var>, <var>m</var>, <var>n</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the quantile (the inverse of the CDF)
at <var>x</var> of the hypergeometric distribution with parameters
<var>t</var>, <var>m</var>, and <var>n</var>.
</p>
<p>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>
<p>The parameters <var>t</var>, <var>m</var>, and <var>n</var> must be positive integers
with <var>m</var> and <var>n</var> not greater than <var>t</var>.
</p></dd></dl>


<a name="XREFkolmogorov_005fsmirnov_005fcdf"></a><dl>
<dt><a name="index-kolmogorov_005fsmirnov_005fcdf"></a>: <em></em> <strong>kolmogorov_smirnov_cdf</strong> <em>(<var>x</var>, <var>tol</var>)</em></dt>
<dd><p>Return the cumulative distribution function (CDF) at <var>x</var> of the
Kolmogorov-Smirnov distribution.
</p>
<p>This is defined as
</p>
<div class="example">
<pre class="example">         Inf
Q(x) =   SUM    (-1)^k exp (-2 k^2 x^2)
       k = -Inf
</pre></div>

<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></dd></dl>


<a name="XREFlaplace_005fpdf"></a><dl>
<dt><a name="index-laplace_005fpdf"></a>: <em></em> <strong>laplace_pdf</strong> <em>(<var>x</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the probability density function (PDF)
at <var>x</var> of the Laplace distribution.
</p></dd></dl>


<a name="XREFlaplace_005fcdf"></a><dl>
<dt><a name="index-laplace_005fcdf"></a>: <em></em> <strong>laplace_cdf</strong> <em>(<var>x</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the cumulative distribution function
(CDF) at <var>x</var> of the Laplace distribution.
</p></dd></dl>


<a name="XREFlaplace_005finv"></a><dl>
<dt><a name="index-laplace_005finv"></a>: <em></em> <strong>laplace_inv</strong> <em>(<var>x</var>)</em></dt>
<dd><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></dd></dl>


<a name="XREFlogistic_005fpdf"></a><dl>
<dt><a name="index-logistic_005fpdf"></a>: <em></em> <strong>logistic_pdf</strong> <em>(<var>x</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the PDF at <var>x</var> of the
logistic distribution.
</p></dd></dl>


<a name="XREFlogistic_005fcdf"></a><dl>
<dt><a name="index-logistic_005fcdf"></a>: <em></em> <strong>logistic_cdf</strong> <em>(<var>x</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the cumulative distribution function
(CDF) at <var>x</var> of the logistic distribution.
</p></dd></dl>


<a name="XREFlogistic_005finv"></a><dl>
<dt><a name="index-logistic_005finv"></a>: <em></em> <strong>logistic_inv</strong> <em>(<var>x</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the quantile (the inverse of the CDF)
at <var>x</var> of the logistic distribution.
</p></dd></dl>


<a name="XREFlognpdf"></a><dl>
<dt><a name="index-lognpdf"></a>: <em></em> <strong>lognpdf</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-lognpdf-1"></a>: <em></em> <strong>lognpdf</strong> <em>(<var>x</var>, <var>mu</var>, <var>sigma</var>)</em></dt>
<dd><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>.
</p>
<p>If a random variable follows this distribution, its logarithm is normally
distributed with mean <var>mu</var> and standard deviation <var>sigma</var>.
</p>
<p>Default values are <var>mu</var> = 0, <var>sigma</var> = 1.
</p></dd></dl>


<a name="XREFlogncdf"></a><dl>
<dt><a name="index-logncdf"></a>: <em></em> <strong>logncdf</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-logncdf-1"></a>: <em></em> <strong>logncdf</strong> <em>(<var>x</var>, <var>mu</var>, <var>sigma</var>)</em></dt>
<dd><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>.
</p>
<p>If a random variable follows this distribution, its logarithm is normally
distributed with mean <var>mu</var> and standard deviation <var>sigma</var>.
</p>
<p>Default values are <var>mu</var> = 0, <var>sigma</var> = 1.
</p></dd></dl>


<a name="XREFlogninv"></a><dl>
<dt><a name="index-logninv"></a>: <em></em> <strong>logninv</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-logninv-1"></a>: <em></em> <strong>logninv</strong> <em>(<var>x</var>, <var>mu</var>, <var>sigma</var>)</em></dt>
<dd><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>.
</p>
<p>If a random variable follows this distribution, its logarithm is normally
distributed with mean <var>mu</var> and standard deviation <var>sigma</var>.
</p>
<p>Default values are <var>mu</var> = 0, <var>sigma</var> = 1.
</p></dd></dl>


<a name="XREFnbinpdf"></a><dl>
<dt><a name="index-nbinpdf"></a>: <em></em> <strong>nbinpdf</strong> <em>(<var>x</var>, <var>n</var>, <var>p</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the probability density function (PDF)
at <var>x</var> of the negative binomial distribution with parameters
<var>n</var> and <var>p</var>.
</p>
<p>When <var>n</var> is integer this is the Pascal distribution.
When <var>n</var> is extended to real numbers this is the Polya distribution.
</p>
<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></dd></dl>


<a name="XREFnbincdf"></a><dl>
<dt><a name="index-nbincdf"></a>: <em></em> <strong>nbincdf</strong> <em>(<var>x</var>, <var>n</var>, <var>p</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the cumulative distribution function
(CDF) at <var>x</var> of the negative binomial distribution with parameters
<var>n</var> and <var>p</var>.
</p>
<p>When <var>n</var> is integer this is the Pascal distribution.
When <var>n</var> is extended to real numbers this is the Polya distribution.
</p>
<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></dd></dl>


<a name="XREFnbininv"></a><dl>
<dt><a name="index-nbininv"></a>: <em></em> <strong>nbininv</strong> <em>(<var>x</var>, <var>n</var>, <var>p</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the quantile (the inverse of the CDF)
at <var>x</var> of the negative binomial distribution with parameters
<var>n</var> and <var>p</var>.
</p>
<p>When <var>n</var> is integer this is the Pascal distribution.
When <var>n</var> is extended to real numbers this is the Polya distribution.
</p>
<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></dd></dl>


<a name="XREFnormpdf"></a><dl>
<dt><a name="index-normpdf"></a>: <em></em> <strong>normpdf</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-normpdf-1"></a>: <em></em> <strong>normpdf</strong> <em>(<var>x</var>, <var>mu</var>, <var>sigma</var>)</em></dt>
<dd><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>mu</var> and
standard deviation <var>sigma</var>.
</p>
<p>Default values are <var>mu</var> = 0, <var>sigma</var> = 1.
</p></dd></dl>


<a name="XREFnormcdf"></a><dl>
<dt><a name="index-normcdf"></a>: <em></em> <strong>normcdf</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-normcdf-1"></a>: <em></em> <strong>normcdf</strong> <em>(<var>x</var>, <var>mu</var>, <var>sigma</var>)</em></dt>
<dd><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>mu</var> and
standard deviation <var>sigma</var>.
</p>
<p>Default values are <var>mu</var> = 0, <var>sigma</var> = 1.
</p></dd></dl>


<a name="XREFnorminv"></a><dl>
<dt><a name="index-norminv"></a>: <em></em> <strong>norminv</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-norminv-1"></a>: <em></em> <strong>norminv</strong> <em>(<var>x</var>, <var>mu</var>, <var>sigma</var>)</em></dt>
<dd><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>mu</var> and
standard deviation <var>sigma</var>.
</p>
<p>Default values are <var>mu</var> = 0, <var>sigma</var> = 1.
</p></dd></dl>


<a name="XREFpoisspdf"></a><dl>
<dt><a name="index-poisspdf"></a>: <em></em> <strong>poisspdf</strong> <em>(<var>x</var>, <var>lambda</var>)</em></dt>
<dd><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></dd></dl>


<a name="XREFpoisscdf"></a><dl>
<dt><a name="index-poisscdf"></a>: <em></em> <strong>poisscdf</strong> <em>(<var>x</var>, <var>lambda</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the cumulative distribution function
(CDF) at <var>x</var> of the Poisson distribution with parameter <var>lambda</var>.
</p></dd></dl>


<a name="XREFpoissinv"></a><dl>
<dt><a name="index-poissinv"></a>: <em></em> <strong>poissinv</strong> <em>(<var>x</var>, <var>lambda</var>)</em></dt>
<dd><p>For each element 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></dd></dl>


<a name="XREFstdnormal_005fpdf"></a><dl>
<dt><a name="index-stdnormal_005fpdf"></a>: <em></em> <strong>stdnormal_pdf</strong> <em>(<var>x</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the probability density function (PDF)
at <var>x</var> of the standard normal distribution
(mean = 0, standard deviation = 1).
</p></dd></dl>


<a name="XREFstdnormal_005fcdf"></a><dl>
<dt><a name="index-stdnormal_005fcdf"></a>: <em></em> <strong>stdnormal_cdf</strong> <em>(<var>x</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the cumulative distribution function
(CDF) at <var>x</var> of the standard normal distribution
(mean = 0, standard deviation = 1).
</p></dd></dl>


<a name="XREFstdnormal_005finv"></a><dl>
<dt><a name="index-stdnormal_005finv"></a>: <em></em> <strong>stdnormal_inv</strong> <em>(<var>x</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the quantile (the
inverse of the CDF) at <var>x</var> of the standard normal distribution
(mean = 0, standard deviation = 1).
</p></dd></dl>


<a name="XREFtpdf"></a><dl>
<dt><a name="index-tpdf"></a>: <em></em> <strong>tpdf</strong> <em>(<var>x</var>, <var>n</var>)</em></dt>
<dd><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></dd></dl>


<a name="XREFtcdf"></a><dl>
<dt><a name="index-tcdf"></a>: <em></em> <strong>tcdf</strong> <em>(<var>x</var>, <var>n</var>)</em></dt>
<dd><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.
</p></dd></dl>


<a name="XREFtinv"></a><dl>
<dt><a name="index-tinv"></a>: <em></em> <strong>tinv</strong> <em>(<var>x</var>, <var>n</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the quantile (the inverse of the CDF)
at <var>x</var> of the t (Student) distribution with <var>n</var>
degrees of freedom.
</p>
<p>This function is analogous to looking in a table for the t-value of a
single-tailed distribution.
</p></dd></dl>


<a name="XREFunidpdf"></a><dl>
<dt><a name="index-unidpdf"></a>: <em></em> <strong>unidpdf</strong> <em>(<var>x</var>, <var>n</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the probability density function (PDF)
at <var>x</var> of a discrete uniform distribution which assumes
the integer values 1&ndash;<var>n</var> with equal probability.
</p>
<p>Warning: The underlying implementation uses the double class and will only
be accurate for <var>n</var> &lt; <code>flintmax</code> (<em>2^{53}</em><!-- /@w --> on
IEEE 754 compatible systems).
</p></dd></dl>


<a name="XREFunidcdf"></a><dl>
<dt><a name="index-unidcdf"></a>: <em></em> <strong>unidcdf</strong> <em>(<var>x</var>, <var>n</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the cumulative distribution function
(CDF) at <var>x</var> of a discrete uniform distribution which assumes
the integer values 1&ndash;<var>n</var> with equal probability.
</p></dd></dl>


<a name="XREFunidinv"></a><dl>
<dt><a name="index-unidinv"></a>: <em></em> <strong>unidinv</strong> <em>(<var>x</var>, <var>n</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the quantile (the inverse of the CDF)
at <var>x</var> of the discrete uniform distribution which assumes
the integer values 1&ndash;<var>n</var> with equal probability.
</p></dd></dl>


<a name="XREFunifpdf"></a><dl>
<dt><a name="index-unifpdf"></a>: <em></em> <strong>unifpdf</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-unifpdf-1"></a>: <em></em> <strong>unifpdf</strong> <em>(<var>x</var>, <var>a</var>, <var>b</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the probability density function (PDF)
at <var>x</var> of the uniform distribution on the interval [<var>a</var>, <var>b</var>].
</p>
<p>Default values are <var>a</var> = 0, <var>b</var> = 1.
</p></dd></dl>


<a name="XREFunifcdf"></a><dl>
<dt><a name="index-unifcdf"></a>: <em></em> <strong>unifcdf</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-unifcdf-1"></a>: <em></em> <strong>unifcdf</strong> <em>(<var>x</var>, <var>a</var>, <var>b</var>)</em></dt>
<dd><p>For each element of <var>x</var>, compute the cumulative distribution function
(CDF) at <var>x</var> of the uniform distribution on the interval
[<var>a</var>, <var>b</var>].
</p>
<p>Default values are <var>a</var> = 0, <var>b</var> = 1.
</p></dd></dl>


<a name="XREFunifinv"></a><dl>
<dt><a name="index-unifinv"></a>: <em></em> <strong>unifinv</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-unifinv-1"></a>: <em></em> <strong>unifinv</strong> <em>(<var>x</var>, <var>a</var>, <var>b</var>)</em></dt>
<dd><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 the interval [<var>a</var>, <var>b</var>].
</p>
<p>Default values are <var>a</var> = 0, <var>b</var> = 1.
</p></dd></dl>


<a name="XREFwblpdf"></a><dl>
<dt><a name="index-wblpdf"></a>: <em></em> <strong>wblpdf</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-wblpdf-1"></a>: <em></em> <strong>wblpdf</strong> <em>(<var>x</var>, <var>scale</var>)</em></dt>
<dt><a name="index-wblpdf-2"></a>: <em></em> <strong>wblpdf</strong> <em>(<var>x</var>, <var>scale</var>, <var>shape</var>)</em></dt>
<dd><p>Compute the probability density function (PDF) at <var>x</var> of the
Weibull distribution with scale parameter <var>scale</var> and
shape parameter <var>shape</var>.
</p>
<p>This is given by
</p>
<div class="example">
<pre class="example">shape * scale^(-shape) * x^(shape-1) * exp (-(x/scale)^shape)
</pre></div>

<p>for <var>x</var> &ge; 0.
</p>
<p>Default values are <var>scale</var> = 1, <var>shape</var> = 1.
</p></dd></dl>


<a name="XREFwblcdf"></a><dl>
<dt><a name="index-wblcdf"></a>: <em></em> <strong>wblcdf</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-wblcdf-1"></a>: <em></em> <strong>wblcdf</strong> <em>(<var>x</var>, <var>scale</var>)</em></dt>
<dt><a name="index-wblcdf-2"></a>: <em></em> <strong>wblcdf</strong> <em>(<var>x</var>, <var>scale</var>, <var>shape</var>)</em></dt>
<dd><p>Compute the cumulative distribution function (CDF) at <var>x</var> of the
Weibull distribution with scale parameter <var>scale</var> and shape
parameter <var>shape</var>.
</p>
<p>This is defined as
</p>
<div class="example">
<pre class="example">1 - exp (-(x/scale)^shape)
</pre></div>

<p>for <var>x</var> &ge; 0.
</p>
<p>Default values are <var>scale</var> = 1, <var>shape</var> = 1.
</p></dd></dl>


<a name="XREFwblinv"></a><dl>
<dt><a name="index-wblinv"></a>: <em></em> <strong>wblinv</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-wblinv-1"></a>: <em></em> <strong>wblinv</strong> <em>(<var>x</var>, <var>scale</var>)</em></dt>
<dt><a name="index-wblinv-2"></a>: <em></em> <strong>wblinv</strong> <em>(<var>x</var>, <var>scale</var>, <var>shape</var>)</em></dt>
<dd><p>Compute the quantile (the inverse of the CDF) at <var>x</var> of the
Weibull distribution with scale parameter <var>scale</var> and
shape parameter <var>shape</var>.
</p>
<p>Default values are <var>scale</var> = 1, <var>shape</var> = 1.
</p></dd></dl>


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