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<a name="Hypergeometric-Random-Variable"></a>
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<p>
Next: <a href="maxima_247.html#Negative-Binomial-Random-Variable" accesskey="n" rel="next">Negative Binomial Random Variable</a>, Previous: <a href="maxima_245.html#Discrete-Uniform-Random-Variable" accesskey="p" rel="previous">Discrete Uniform Random Variable</a>, Up: <a href="maxima_239.html#Functions-and-Variables-for-discrete-distributions" accesskey="u" rel="up">Functions and Variables for discrete 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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<a name="Hypergeometric-Random-Variable-1"></a>
<h4 class="subsection">52.3.7 Hypergeometric Random Variable</h4>
<p>The <em>hypergeometric distribution</em> is a discrete probability
distribution.
</p>
<p>Let <em>n_1</em> be the number of objects of a class
<em>A</em> and <em>n_2</em> be the number of objects of class <em>B</em>.
We take out <em>n</em> objects, <em>without</em> replacment. Then the
hypergeometric distribution is the probability that exactly <em>k</em>
objects are from class <em>A</em>. Of course <em>n \leq n_1 + n_2</em>.
</p>
<a name="pdf_005fhypergeometric"></a><a name="Item_003a-distrib_002fdeffn_002fpdf_005fhypergeometric"></a><dl>
<dt><a name="index-pdf_005fhypergeometric"></a>Function: <strong>pdf_hypergeometric</strong> <em>(<var>x</var>,<var>n_1</var>,<var>n_2</var>,<var>n</var>)</em></dt>
<dd><p>Returns the value at <var>x</var> of the probability function of a
\({\it Hypergeometric}(n1,n2,n)\)</p>
<p>random variable, with <em>n_1</em>, <em>n_2</em> and <em>n</em> non negative
integers and <em>n\leq n_1+n_2</em>.
Being <em>n_1</em> the number of objects of class A, <em>n_2</em> the number of objects of class B, and
<em>n</em> the size of the sample without replacement, this function returns the probability of
event "exactly <var>x</var> objects are of class A".
</p>
<p>To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The pdf is
$$
f(x; n_1, n_2, n) = {\displaystyle{n_1\choose x} {n_2 \choose n-x}
\over \displaystyle{n_2+n_1 \choose n}}
$$</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="cdf_005fhypergeometric"></a><a name="Item_003a-distrib_002fdeffn_002fcdf_005fhypergeometric"></a><dl>
<dt><a name="index-cdf_005fhypergeometric"></a>Function: <strong>cdf_hypergeometric</strong> <em>(<var>x</var>,<var>n_1</var>,<var>n_2</var>,<var>n</var>)</em></dt>
<dd><p>Returns the value at <var>x</var> of the distribution function of a
\({\it Hypergeometric}(n1,n2,n)\)</p>
<p>random variable, with <em>n_1</em>, <em>n_2</em> and <em>n</em> non negative
integers and <em>n\leq n_1+n_2</em>.
See <code>pdf_hypergeometric</code> for a more complete description.
</p>
<p>To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The cdf is
$$
F(x; n_1, n_2, n) = {n_2+n_1\choose n}^{-1}
\sum_{k=0}^{\lfloor x \rfloor} {n_1 \choose k} {n_2 \choose n - k}
$$</p>
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</dd></dl>
<a name="quantile_005fhypergeometric"></a><a name="Item_003a-distrib_002fdeffn_002fquantile_005fhypergeometric"></a><dl>
<dt><a name="index-quantile_005fhypergeometric"></a>Function: <strong>quantile_hypergeometric</strong> <em>(<var>q</var>,<var>n1</var>,<var>n2</var>,<var>n</var>)</em></dt>
<dd><p>Returns the <var>q</var>-quantile of a
\({\it Hypergeometric}(n1,n2,n)\) random
variable, with <var>n1</var>, <var>n2</var> and <var>n</var> non negative integers
and <em>n\leq n1+n2</em>; in other words, this is the inverse of <code>cdf_hypergeometric</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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</dd></dl>
<a name="mean_005fhypergeometric"></a><a name="Item_003a-distrib_002fdeffn_002fmean_005fhypergeometric"></a><dl>
<dt><a name="index-mean_005fhypergeometric"></a>Function: <strong>mean_hypergeometric</strong> <em>(<var>n_1</var>,<var>n_2</var>,<var>n</var>)</em></dt>
<dd><p>Returns the mean of a discrete uniform random variable
\({\it Hypergeometric}(n_1,n_2,n)\), with <em>n_1</em>, <em>n_2</em> and <em>n</em> non negative integers and <em>n\leq n_1+n_2</em>. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The mean is
$$
E[X] = {n n_1\over n_2+n_1}
$$</p>
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</dd></dl>
<a name="var_005fhypergeometric"></a><a name="Item_003a-distrib_002fdeffn_002fvar_005fhypergeometric"></a><dl>
<dt><a name="index-var_005fhypergeometric"></a>Function: <strong>var_hypergeometric</strong> <em>(<var>n1</var>,<var>n2</var>,<var>n</var>)</em></dt>
<dd><p>Returns the variance of a hypergeometric random variable
\({\it Hypergeometric}(n_1,n_2,n)\), with <em>n1</em>, <em>n2</em> and <em>n</em> non negative integers and <em>n<=n1+n2</em>. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The variance is
$$
V[X] = {n n_1 n_2 (n_1 + n_2 - n)
\over
(n_1 + n_2 - 1) (n_1 + n_2)^2}
$$</p>
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</dd></dl>
<a name="std_005fhypergeometric"></a><a name="Item_003a-distrib_002fdeffn_002fstd_005fhypergeometric"></a><dl>
<dt><a name="index-std_005fhypergeometric"></a>Function: <strong>std_hypergeometric</strong> <em>(<var>n_1</var>,<var>n_2</var>,<var>n</var>)</em></dt>
<dd><p>Returns the standard deviation of a
\({\it Hypergeometric}(n_1,n_2,n)\) random variable, with <em>n_1</em>, <em>n_2</em> and <em>n</em> non negative integers and <em>n\leq n_1+n_2</em>. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The standard deviation is
$$
D[X] = {1\over n_1+n_2}\sqrt{n n_1 n_2 (n_1 + n_2 - n) \over n_1+n_2-1}
$$</p>
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</dd></dl>
<a name="skewness_005fhypergeometric"></a><a name="Item_003a-distrib_002fdeffn_002fskewness_005fhypergeometric"></a><dl>
<dt><a name="index-skewness_005fhypergeometric"></a>Function: <strong>skewness_hypergeometric</strong> <em>(<var>n_1</var>,<var>n_2</var>,<var>n</var>)</em></dt>
<dd><p>Returns the skewness coefficient of a
\({\it Hypergeometric}(n1,n2,n)\) random variable, with <em>n_1</em>, <em>n_2</em> and <em>n</em> non negative integers and <em>n\leq n1+n2</em>. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The skewness coefficient is
$$
SK[X] = {(n_2-n_2)(n_1+n_2-2n)\over n_1+n_2-2}
\sqrt{n_1+n_2-1 \over n n_1 n_2 (n_1+n_2-n)}
$$</p>
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</dd></dl>
<a name="kurtosis_005fhypergeometric"></a><a name="Item_003a-distrib_002fdeffn_002fkurtosis_005fhypergeometric"></a><dl>
<dt><a name="index-kurtosis_005fhypergeometric"></a>Function: <strong>kurtosis_hypergeometric</strong> <em>(<var>n_1</var>,<var>n_2</var>,<var>n</var>)</em></dt>
<dd><p>Returns the kurtosis coefficient of a
\({\it Hypergeometric}(n_1,n_2,n)\) random variable, with <em>n_1</em>, <em>n_2</em> and <em>n</em> non negative integers and <em>n\leq n1+n2</em>. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The kurtosis coefficient is
$$
\eqalign{
KU[X] = &
\left[{C(1)C(0)^2
\over
n n_1 n_2 C(3)C(2)C(n)}\right. \cr
& \times
\left.\left(
{3n_1n_2\left((n-2)C(0)^2+6nC(n)-n^2C(0)\right)
\over
C(0)^2
}
-6nC(n) + C(0)C(-1)
\right)\right] \cr
&-3
}
$$</p>
<p>where
\(C(k) = n_1+n_2-k\).
</p>
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Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="random_005fhypergeometric"></a><a name="Item_003a-distrib_002fdeffn_002frandom_005fhypergeometric"></a><dl>
<dt><a name="index-random_005fhypergeometric"></a>Function: <strong>random_hypergeometric</strong> <em>(<var>n1</var>,<var>n2</var>,<var>n</var>) <br> <tt>random_hypergeometric</tt> (<var>n1</var>,<var>n2</var>,<var>n</var>,<var>m</var>)</em></dt>
<dd>
<p>Returns a
\({\it Hypergeometric}(n1,n2,n)\) random variate, with <var>n1</var>, <var>n2</var> and <var>n</var> non negative integers and <em>n<=n1+n2</em>. Calling <code>random_hypergeometric</code> with a fourth argument <var>m</var>, a random sample of size <var>m</var> will be simulated.
</p>
<p>Algorithm described in Kachitvichyanukul, V., Schmeiser, B.W. (1985) <var>Computer generation of hypergeometric random variates.</var> Journal of Statistical Computation and Simulation 22, 127-145.
</p>
<p>To make use of this function, write first <code>load("distrib")</code>.
</p>
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·<a href="maxima_424.html#Category_003a-Random-numbers">Random numbers</a>
·</div>
</dd></dl>
<a name="Item_003a-distrib_002fnode_002fNegative-Binomial-Random-Variable"></a><hr>
<div class="header">
<p>
Next: <a href="maxima_247.html#Negative-Binomial-Random-Variable" accesskey="n" rel="next">Negative Binomial Random Variable</a>, Previous: <a href="maxima_245.html#Discrete-Uniform-Random-Variable" accesskey="p" rel="previous">Discrete Uniform Random Variable</a>, Up: <a href="maxima_239.html#Functions-and-Variables-for-discrete-distributions" accesskey="u" rel="up">Functions and Variables for discrete 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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