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<a name="Gamma-Random-Variable"></a>
<div class="header">
<p>
Next: <a href="maxima_230.html#Beta-Random-Variable" accesskey="n" rel="next">Beta Random Variable</a>, Previous: <a href="maxima_228.html#Lognormal-Random-Variable" accesskey="p" rel="previous">Lognormal Random Variable</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="Gamma-Random-Variable-1"></a>
<h4 class="subsection">52.2.9 Gamma Random Variable</h4>
<p>The <em>gamma distribution</em> is a two-parameter family of probability
distributions. Maxima uses the parameterization using the shape and
scale for the first and second parameters of the distribution.
</p>
<a name="pdf_005fgamma"></a><a name="Item_003a-distrib_002fdeffn_002fpdf_005fgamma"></a><dl>
<dt><a name="index-pdf_005fgamma"></a>Function: <strong>pdf_gamma</strong> <em>(<var>x</var>,<var>a</var>,<var>b</var>)</em></dt>
<dd><p>Returns the value at <var>x</var> of the density function of a
\(\Gamma\left(a,b\right)\) random variable, with <em>a,b>0</em>. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The shape parameter is <em>a</em>, and the scale parameter is <em>b</em>.
</p>
<p>The pdf is
$$
f(x; a, b) = {x^{a-1}e^{-x/b}\over b^a \Gamma(a)}
$$</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="cdf_005fgamma"></a><a name="Item_003a-distrib_002fdeffn_002fcdf_005fgamma"></a><dl>
<dt><a name="index-cdf_005fgamma"></a>Function: <strong>cdf_gamma</strong> <em>(<var>x</var>,<var>a</var>,<var>b</var>)</em></dt>
<dd><p>Returns the value at <var>x</var> of the distribution function of a
\(\Gamma\left(a,b\right)\) random variable, with <em>a,b>0</em>.
</p>
<p>The cdf is
$$
F(x; a, b) =
\cases{
1-Q(a,{x\over b}) & for $x \ge 0$ \cr
\cr
0 & for $x < 0$
}
$$</p>
<p>where <em>Q(a,z)</em> is the <a href="maxima_87.html#gamma_005fincomplete_005fregularized">gamma_incomplete_regularized</a> function.
</p><div class="example">
<pre class="example">(%i1) load ("distrib")$
</pre><pre class="example">(%i2) cdf_gamma(3,5,21);
1
(%o2) 1 - gamma_incomplete_regularized(5, -)
7
</pre><pre class="example">(%i3) float(%);
(%o3) 4.402663157376807e-7
</pre></div>
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Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
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</dd></dl>
<a name="quantile_005fgamma"></a><a name="Item_003a-distrib_002fdeffn_002fquantile_005fgamma"></a><dl>
<dt><a name="index-quantile_005fgamma"></a>Function: <strong>quantile_gamma</strong> <em>(<var>q</var>,<var>a</var>,<var>b</var>)</em></dt>
<dd><p>Returns the <var>q</var>-quantile of a
\(\Gamma\left(a,b\right)\) random variable, with <em>a,b>0</em>; in other words, this is the inverse of <code>cdf_gamma</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>
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Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
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</dd></dl>
<a name="mean_005fgamma"></a><a name="Item_003a-distrib_002fdeffn_002fmean_005fgamma"></a><dl>
<dt><a name="index-mean_005fgamma"></a>Function: <strong>mean_gamma</strong> <em>(<var>a</var>,<var>b</var>)</em></dt>
<dd><p>Returns the mean of a
\(\Gamma\left(a,b\right)\) random variable, with <em>a,b>0</em>. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The mean is
$$
E[X] = ab
$$</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="var_005fgamma"></a><a name="Item_003a-distrib_002fdeffn_002fvar_005fgamma"></a><dl>
<dt><a name="index-var_005fgamma"></a>Function: <strong>var_gamma</strong> <em>(<var>a</var>,<var>b</var>)</em></dt>
<dd><p>Returns the variance of a
\(\Gamma\left(a,b\right)\) random variable, with <em>a,b>0</em>. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The variance is
$$
V[X] = ab^2
$$</p>
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Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="std_005fgamma"></a><a name="Item_003a-distrib_002fdeffn_002fstd_005fgamma"></a><dl>
<dt><a name="index-std_005fgamma"></a>Function: <strong>std_gamma</strong> <em>(<var>a</var>,<var>b</var>)</em></dt>
<dd><p>Returns the standard deviation of a
\(\Gamma\left(a,b\right)\) random variable, with <em>a,b>0</em>. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The standard deviation is
$$
D[X] = b\sqrt{a}
$$</p>
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Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="skewness_005fgamma"></a><a name="Item_003a-distrib_002fdeffn_002fskewness_005fgamma"></a><dl>
<dt><a name="index-skewness_005fgamma"></a>Function: <strong>skewness_gamma</strong> <em>(<var>a</var>,<var>b</var>)</em></dt>
<dd><p>Returns the skewness coefficient of a
\(\Gamma\left(a,b\right)\) random variable, with <em>a,b>0</em>. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The skewness coefficient is
$$
SK[X] = {2\over \sqrt{a}}
$$</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="kurtosis_005fgamma"></a><a name="Item_003a-distrib_002fdeffn_002fkurtosis_005fgamma"></a><dl>
<dt><a name="index-kurtosis_005fgamma"></a>Function: <strong>kurtosis_gamma</strong> <em>(<var>a</var>,<var>b</var>)</em></dt>
<dd><p>Returns the kurtosis coefficient of a
\(\Gamma\left(a,b\right)\) random variable, with <em>a,b>0</em>. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The kurtosis coefficient is
$$
KU[X] = {6\over a}
$$</p>
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Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="random_005fgamma"></a><a name="Item_003a-distrib_002fdeffn_002frandom_005fgamma"></a><dl>
<dt><a name="index-random_005fgamma"></a>Function: <strong>random_gamma</strong> <em>(<var>a</var>,<var>b</var>) <br> <tt>random_gamma</tt> (<var>a</var>,<var>b</var>,<var>n</var>)</em></dt>
<dd>
<p>Returns a
\(\Gamma\left(a,b\right)\) random variate, with <em>a,b>0</em>. Calling <code>random_gamma</code> with a third argument <var>n</var>, a random sample of size <var>n</var> will be simulated.
</p>
<p>The implemented algorithm is a combination of two procedures, depending on the value of parameter <var>a</var>:
</p>
<p>For <em>a>=1</em>, Cheng, R.C.H. and Feast, G.M. (1979). <var>Some simple gamma variate generators</var>. Appl. Stat., 28, 3, 290-295.
</p>
<p>For <em>0<a<1</em>, Ahrens, J.H. and Dieter, U. (1974). <var>Computer methods for sampling from gamma, , poisson and binomial cdf_tributions</var>. Computing, 12, 223-246.
</p>
<p>To make use of this function, write first <code>load("distrib")</code>.
</p>
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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_002fBeta-Random-Variable"></a><hr>
<div class="header">
<p>
Next: <a href="maxima_230.html#Beta-Random-Variable" accesskey="n" rel="next">Beta Random Variable</a>, Previous: <a href="maxima_228.html#Lognormal-Random-Variable" accesskey="p" rel="previous">Lognormal Random Variable</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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