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<a name="rmath-statistical-functions"></a>
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<hr size="1">
<a name="rmath-statistical-functions-1"></a>
<h3 class="subsection">D.16.1 <code>rmath</code> statistical functions</h3>

<p>Functions for evaluating random draws, density, cumulative probability
and inverse cumulative probability are provided for some of the more
frequently used probability distributions, which are chi-squared,
non-central chi-squared, exponential, lognormal, normal, poisson,
Student&rsquo;s t, and uniform.
</p>
<p>Each distribution is known by an abbreviated name and specified by one
<a name="index-distributions"></a>
<a name="index-probability-distributions"></a>
<a name="index-statistical-distributions"></a>
or two real parameters as listed below. Names of distributions in this
table form the stem of a library function name.  The names of the
parameters such as <var>mu</var> and <var>sigma</var> are not explicitly
mentioned when invoking the functions, but are listed here for
reference.  The precise definitions of the distribution functions and
interpretations of these parameters can be found in standard texts on
probability and statistics.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">chisq         <var>df</var>
nchisq        <var>df</var>, <var>lambda</var>
exp           <var>scale</var>
lnorm         <var>logmean</var>, <var>logsd</var>
norm          <var>mu</var>, <var>sigma</var>
pois          <var>lambda</var>
t             <var>n</var>
unif          <var>a</var>, <var>b</var>
</pre></td></tr></table>

<p>The virtual code interface follows a naming convention similar to the
native API, in that function names beginning with <code>r</code> represent
random draws from a distribution, with the argument to the function
being the parameters specifying the distribution. Functions in this
first group return a random draw from a distribution described by a
single real parameter.
</p>
<ul>
<li> <code>rchisq</code>
</li><li> <code>rexp</code>
</li><li> <code>rpois</code>
</li><li> <code>rt</code>
</li></ul>

<p>These next functions return random draws from distributions specified
by a pair of parameters, <code>(<var>x</var>,<var>y</var>)</code>.
</p>
<ul>
<li> <code>rnchisq</code>
</li><li> <code>rlnorm</code>
</li><li> <code>rnorm</code>
</li><li> <code>runif</code>
</li></ul>

<p>Functions whose names begin with <code>d</code> evaluate the probability
density of a distribution at a given point. They require at least two
real arguments, the first being the point whose probability density is
sought, and the remaining ones being the parameters that specify the
distribution. A boolean operand, which is <code>nil</code> for false and
<code>(nil,nil)</code> for true, requests the logarithm of the density when
true.
</p>
<p>Functions with names in the following group take a triple with two
real operands and a boolean, <code>(<var>x</var>,(<var>y</var>,<var>a</var>))</code>, and
return a probabiity density.
</p>
<ul>
<li> <code>dchisq</code>
</li><li> <code>dexp</code>
</li><li> <code>dpois</code>
</li><li> <code>dt</code>
</li></ul>

<p>The next functions pertain to distributions requiring two paramters to
specify them, so they take a quadruple with three real operands and a
boolean, <code>(<var>x</var>,(<var>y</var>,(<var>z</var>,<var>a</var>)))</code>.
</p>
<ul>
<li> <code>dnchisq</code>
</li><li> <code>dlnorm</code>
</li><li> <code>dnorm</code>
</li><li> <code>dunif</code>
</li></ul>

<p>Functions whose names begin with <code>p</code> or <code>q</code> obtain
<a name="index-cumulative-probability"></a>
cumulative probabilities or inverse cumulative probabilities
respectively for a specified distribution. They require one real
operand to identify the point whose probability or inverse probability
is sought, and other real operands to parameterize the distribution,
as above. There are also two boolean operands. The first is true in
order to request a probability or inverse probability with respect to
the lower tail as opposed to the upper, and the other is true to
indicate that probabilities are to be expressed logarithmically.
</p>
<p>The argument to these functions is a quadruple with two real operands
and two booleans, <code>(<var>x</var>,(<var>y</var>,(<var>a</var>,<var>b</var>)))</code>.
</p>
<ul>
<li> <code>pchisq</code>, <code>qchisq</code>
</li><li> <code>pexp</code>, <code>qexp</code>
</li><li> <code>ppois</code>, <code>qpois</code>
</li><li> <code>pt</code>, <code>qt</code>
</li></ul>

<p>The remaining functions pertain to distributions parameterized by two
real operands. These take a quintuple with three real operands and two
booleans, <code>(<var>x</var>,(<var>y</var>,(<var>z</var>,(<var>a</var>,<var>b</var>))))</code>.
</p>
<ul>
<li> <code>pnchisq</code>,
<code>qnchisq</code>
</li><li> <code>plnorm</code>,
<code>qlnorm</code>
</li><li> <code>pnorm</code>,
<code>qnorm</code>
</li><li> <code>punif</code>,
<code>qunif</code>
</li></ul>

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