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<table width="100%" summary="page for GammaDist {stats}"><tr><td>GammaDist {stats}</td><td style="text-align: right;">R Documentation</td></tr></table>

<h2>The Gamma Distribution</h2>

<h3>Description</h3>

Density, distribution function, quantile function and random
generation for the Gamma distribution with parameters <code>shape</code> and
<code>scale</code>.

<h3>Usage</h3>

<pre>
dgamma(x, shape, rate = 1, scale = 1/rate, log = FALSE)
pgamma(q, shape, rate = 1, scale = 1/rate, lower.tail = TRUE,
 log.p = FALSE)
qgamma(p, shape, rate = 1, scale = 1/rate, lower.tail = TRUE,
 log.p = FALSE)
rgamma(n, shape, rate = 1, scale = 1/rate)
</pre>

<h3>Arguments</h3>

<table summary="R argblock">
<tr valign="top"><td><code>x, q</code></td>
<td>
vector of quantiles.
</td></tr>
<tr valign="top"><td><code>p</code></td>
<td>
vector of probabilities.
</td></tr>
<tr valign="top"><td><code>n</code></td>
<td>
number of observations. If <code>length(n) &gt; 1</code>, the length
is taken to be the number required.
</td></tr>
<tr valign="top"><td><code>rate</code></td>
<td>
an alternative way to specify the scale.
</td></tr>
<tr valign="top"><td><code>shape, scale</code></td>
<td>
shape and scale parameters. Must be positive,
<code>scale</code> strictly.
</td></tr>
<tr valign="top"><td><code>log, log.p</code></td>
<td>
logical; if <code>TRUE</code>, probabilities/densities p
are returned as log(p).
</td></tr>
<tr valign="top"><td><code>lower.tail</code></td>
<td>
logical; if TRUE (default), probabilities are
P[X &le; x], otherwise, P[X &gt; x].
</td></tr>
</table>

<h3>Details</h3>

If <code>scale</code> is omitted, it assumes the default value of <code>1</code>.

The Gamma distribution with parameters <code>shape</code> = a
and <code>scale</code> = s has density

f(x)= 1/(s^a Gamma(a)) x^(a-1) e^-(x/s)

for x &ge; 0, a &gt; 0 and s &gt; 0.
(Here Gamma(a) is the function implemented by R's
<code><a href="../../base/html/Special.html">gamma</a>()</code> and defined in its help. Note that a = 0
corresponds to the trivial distribution with all mass at point 0.)

The mean and variance are
E(X) = a*s and
Var(X) = a*s^2.

The cumulative hazard H(t) = - log(1 - F(t))
is

<pre>-pgamma(t, ..., lower = FALSE, log = TRUE)
</pre>
Note that for smallish values of <code>shape</code> (and moderate
<code>scale</code>) a large parts of the mass of the Gamma distribution is
on values of x so near zero that they will be represented as
zero in computer arithmetic. So <code>rgamma</code> may well return values
which will be represented as zero. (This will also happen for very
large values of <code>scale</code> since the actual generation is done for
<code>scale = 1</code>.)

<h3>Value</h3>

<code>dgamma</code> gives the density,
<code>pgamma</code> gives the distribution function,
<code>qgamma</code> gives the quantile function, and
<code>rgamma</code> generates random deviates.

Invalid arguments will result in return value <code>NaN</code>, with a warning.

The length of the result is determined by <code>n</code> for
<code>rgamma</code>, and is the maximum of the lengths of the
numerical arguments for the other functions. 

The numerical arguments other than <code>n</code> are recycled to the
length of the result. Only the first elements of the logical
arguments are used.

The S (Becker et al, 1988) parametrization was via <code>shape</code>
and <code>rate</code>: S had no <code>scale</code> parameter. It is an error
to supply and <code>scale</code> and <code>rate</code>.

<code>pgamma</code> is closely related to the incomplete gamma function. As
defined by Abramowitz and Stegun 6.5.1 (and by &lsquo;Numerical
Recipes&rsquo;) this is

P(a,x) = 1/Gamma(a) integral_0^x t^(a-1) exp(-t) dt

P(a, x) is <code>pgamma(x, a)</code>. Other authors (for example
Karl Pearson in his 1922 tables) omit the normalizing factor,
defining the incomplete gamma function &gamma;(a,x) as
gamma(a,x) =
 integral_0^x t^(a-1) exp(-t) dt, i.e., <code>pgamma(x, a) * gamma(a)</code>.
Yet other use the &lsquo;upper&rsquo; incomplete gamma function,

Gamma(a,x) = integral_x^Inf t^(a-1) exp(-t) dt,

which can be computed by
<code>pgamma(x, a, lower = FALSE) * gamma(a)</code>.

Note however that <code>pgamma(x, a, ..)</code> currently requires a &gt; 0,
whereas the incomplete gamma function is also defined for negative
a. In that case, you can use <code>gamma_inc(a,x)</code> (for
&Gamma;(a,x)) from package <a href="https://CRAN.R-project.org/package=gsl">gsl</a>.

See also
<a href="https://en.wikipedia.org/wiki/Incomplete_gamma_function">https://en.wikipedia.org/wiki/Incomplete_gamma_function</a>, or
<a href="http://dlmf.nist.gov/8.2#i">http://dlmf.nist.gov/8.2#i</a>.

<h3>Source</h3>

<code>dgamma</code> is computed via the Poisson density, using code contributed
by Catherine Loader (see <code><a href="Binomial.html">dbinom</a></code>).

<code>pgamma</code> uses an unpublished (and not otherwise documented)
algorithm &lsquo;mainly by Morten Welinder&rsquo;.

<code>qgamma</code> is based on a C translation of

Best, D. J. and D. E. Roberts (1975).
Algorithm AS91. Percentage points of the chi-squared distribution.
Applied Statistics, 24, 385&ndash;388.

plus a final Newton step to improve the approximation.

<code>rgamma</code> for <code>shape &gt;= 1</code> uses

Ahrens, J. H. and Dieter, U. (1982).
Generating gamma variates by a modified rejection technique.
Communications of the ACM, 25, 47&ndash;54,

and for <code>0 &lt; shape &lt; 1</code> uses

Ahrens, J. H. and Dieter, U. (1974).
Computer methods for sampling from gamma, beta, Poisson and binomial
distributions. Computing, 12, 223&ndash;246.

<h3>References</h3>

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988).
The New S Language.
Wadsworth &amp; Brooks/Cole.

Shea, B. L. (1988).
Algorithm AS 239: Chi-squared and incomplete Gamma integral,
Applied Statistics (JRSS C), 37, 466&ndash;473.
doi: <a href="https://doi.org/10.2307/2347328">10.2307/2347328</a>.

Abramowitz, M. and Stegun, I. A. (1972)
Handbook of Mathematical Functions. New York: Dover.
Chapter 6: Gamma and Related Functions.

NIST Digital Library of Mathematical Functions.
<a href="http://dlmf.nist.gov/">http://dlmf.nist.gov/</a>, section 8.2.

<code><a href="../../base/html/Special.html">gamma</a></code> for the gamma function.

<a href="Distributions.html">Distributions</a> for other standard distributions, including
<code><a href="Beta.html">dbeta</a></code> for the Beta distribution and <code><a href="Chisquare.html">dchisq</a></code>
for the chi-squared distribution which is a special case of the Gamma
distribution.

<h3>Examples</h3>

<pre>
-log(dgamma(1:4, shape = 1))
p &lt;- (1:9)/10
pgamma(qgamma(p, shape = 2), shape = 2)
1 - 1/exp(qgamma(p, shape = 1))

# even for shape = 0.001 about half the mass is on numbers
# that cannot be represented accurately (and most of those as zero)
pgamma(.Machine$double.xmin, 0.001)
pgamma(5e-324, 0.001) # on most machines 5e-324 is the smallest
 # representable non-zero number
table(rgamma(1e4, 0.001) == 0)/1e4
</pre>

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