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<a name="Binomial-Random-Variable"></a>
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<p>
Next: <a href="maxima_242.html#Poisson-Random-Variable" accesskey="n" rel="next">Poisson Random Variable</a>, Previous: <a href="maxima_240.html#General-Finite-Discrete-Random-Variable" accesskey="p" rel="previous">General Finite Discrete 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>
</div>
<a name="Binomial-Random-Variable-1"></a>
<h4 class="subsection">52.3.2 Binomial Random Variable</h4>
<p>The <em>binomial distribution</em> with parameters <em>n</em> and <em>p</em>
is a discrete probability distribution. It consists of <em>n</em>
independent experiments where each experiment consists of a
Boolean-valued outcome where a success occurs with a probablity
<em>p</em>.
</p>
<p>For example, a biased coin that comes up heads with probablity
<em>p</em> is tossed <em>n</em> times. Then the probability of exactly
<em>k</em> heads in <em>n</em> tosses is given by the binomial
distribution.
</p>
<a name="pdf_005fbinomial"></a><a name="Item_003a-distrib_002fdeffn_002fpdf_005fbinomial"></a><dl>
<dt><a name="index-pdf_005fbinomial"></a>Function: <strong>pdf_binomial</strong> <em>(<var>x</var>,<var>n</var>,<var>p</var>)</em></dt>
<dd><p>Returns the value at <var>x</var> of the probability function of a
\({\it Binomial}(n,p)\) random variable, with <em>0 \leq p \leq 1</em> and <em>n</em> a positive integer. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The pdf is
$$
f(x; n, p) = {n\choose x} (1-p)^{n-x}p^x
$$</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="cdf_005fbinomial"></a><a name="Item_003a-distrib_002fdeffn_002fcdf_005fbinomial"></a><dl>
<dt><a name="index-cdf_005fbinomial"></a>Function: <strong>cdf_binomial</strong> <em>(<var>x</var>,<var>n</var>,<var>p</var>)</em></dt>
<dd><p>Returns the value at <var>x</var> of the distribution function of a
\({\it Binomial}(n,p)\) random variable, with <em>0 \leq p \leq 1</em> and <em>n</em> a positive integer.
</p>
<p>The cdf is
$$
F(x; n, p) = I_{1-p}(n-\lfloor x \rfloor, \lfloor x \rfloor + 1)
$$</p>
<p>where
\(I_z(a,b)\) is the <a href="maxima_87.html#beta_005fincomplete_005fregularized">beta_incomplete_regularized</a>
function.
</p>
<div class="example">
<pre class="example">(%i1) load ("distrib")$
</pre><pre class="example">(%i2) cdf_binomial(5,7,1/6);
7775
(%o2) ----
7776
</pre><pre class="example">(%i3) float(%);
(%o3) 0.9998713991769548
</pre></div>
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Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="quantile_005fbinomial"></a><a name="Item_003a-distrib_002fdeffn_002fquantile_005fbinomial"></a><dl>
<dt><a name="index-quantile_005fbinomial"></a>Function: <strong>quantile_binomial</strong> <em>(<var>q</var>,<var>n</var>,<var>p</var>)</em></dt>
<dd><p>Returns the <var>q</var>-quantile of a
\({\it Binomial}(n,p)\) random variable, with <em>0 \leq p \leq 1</em> and <em>n</em> a positive integer; in other words, this is the inverse of <code>cdf_binomial</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>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="mean_005fbinomial"></a><a name="Item_003a-distrib_002fdeffn_002fmean_005fbinomial"></a><dl>
<dt><a name="index-mean_005fbinomial"></a>Function: <strong>mean_binomial</strong> <em>(<var>n</var>,<var>p</var>)</em></dt>
<dd><p>Returns the mean of a
\({\it Binomial}(n,p)\) random variable, with <em>0 \leq p \leq 1</em> and <em>n</em> a positive integer. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The mean is
$$
E[X] = np
$$</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="var_005fbinomial"></a><a name="Item_003a-distrib_002fdeffn_002fvar_005fbinomial"></a><dl>
<dt><a name="index-var_005fbinomial"></a>Function: <strong>var_binomial</strong> <em>(<var>n</var>,<var>p</var>)</em></dt>
<dd><p>Returns the variance of a
\({\it Binomial}(n,p)\) random variable, with <em>0 \leq p \leq 1</em> and <em>n</em> a positive integer. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The variance is
$$
V[X] = np(1-p)
$$</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="std_005fbinomial"></a><a name="Item_003a-distrib_002fdeffn_002fstd_005fbinomial"></a><dl>
<dt><a name="index-std_005fbinomial"></a>Function: <strong>std_binomial</strong> <em>(<var>n</var>,<var>p</var>)</em></dt>
<dd><p>Returns the standard deviation of a
\({\it Binomial}(n,p)\) random variable, with <em>0 \leq p \leq 1</em> and <em>n</em> a positive integer. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The standard deviation is
$$
D[X] = \sqrt{np(1-p)}
$$</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="skewness_005fbinomial"></a><a name="Item_003a-distrib_002fdeffn_002fskewness_005fbinomial"></a><dl>
<dt><a name="index-skewness_005fbinomial"></a>Function: <strong>skewness_binomial</strong> <em>(<var>n</var>,<var>p</var>)</em></dt>
<dd><p>Returns the skewness coefficient of a
\({\it Binomial}(n,p)\) random variable, with <em>0 \leq p \leq 1</em> and <em>n</em> a positive integer. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The skewness coefficient is
$$
SK[X] = {1-2p\over \sqrt{np(1-p)}}
$$</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="kurtosis_005fbinomial"></a><a name="Item_003a-distrib_002fdeffn_002fkurtosis_005fbinomial"></a><dl>
<dt><a name="index-kurtosis_005fbinomial"></a>Function: <strong>kurtosis_binomial</strong> <em>(<var>n</var>,<var>p</var>)</em></dt>
<dd><p>Returns the kurtosis coefficient of a
\({\it Binomial}(n,p)\) random variable, with <em>0 \leq p \leq 1</em> and <em>n</em> a positive integer. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The kurtosis coefficient is
$$
KU[X] = {1-6p(1-p)\over np(1-p)}
$$</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="random_005fbinomial"></a><a name="Item_003a-distrib_002fdeffn_002frandom_005fbinomial"></a><dl>
<dt><a name="index-random_005fbinomial"></a>Function: <strong>random_binomial</strong> <em>(<var>n</var>,<var>p</var>) <br> <tt>random_binomial</tt> (<var>n</var>,<var>p</var>,<var>m</var>)</em></dt>
<dd>
<p>Returns a
\({\it Binomial}(n,p)\) random variate, with <em>0 \leq p \leq 1</em> and <em>n</em> a positive integer. Calling <code>random_binomial</code> with a third argument <var>m</var>, a random sample of size <var>m</var> will be simulated.
</p>
<p>The implemented algorithm is based on the one described in Kachitvichyanukul, V. and Schmeiser, B.W. (1988) <var>Binomial Random Variate Generation</var>. Communications of the ACM, 31, Feb., 216.
</p>
<p>To make use of this function, write first <code>load("distrib")</code>.
</p>
<div class=categorybox>
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_002fPoisson-Random-Variable"></a><hr>
<div class="header">
<p>
Next: <a href="maxima_242.html#Poisson-Random-Variable" accesskey="n" rel="next">Poisson Random Variable</a>, Previous: <a href="maxima_240.html#General-Finite-Discrete-Random-Variable" accesskey="p" rel="previous">General Finite Discrete 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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