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<a name="Geometric-Random-Variable"></a>
<div class="header">
<p>
Next: <a href="maxima_245.html#Discrete-Uniform-Random-Variable" accesskey="n" rel="next">Discrete Uniform Random Variable</a>, Previous: <a href="maxima_243.html#Bernoulli-Random-Variable" accesskey="p" rel="previous">Bernoulli 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="Geometric-Random-Variable-1"></a>
<h4 class="subsection">52.3.5 Geometric Random Variable</h4>
<p>The <em>Geometric distibution</em> is a discrete probability
distribution. It is the distribution of the number
Bernoulli trials that fail before the first success.
</p>
<p>Consider flipping a biased coin where heads occurs with probablity
<em>p</em>. Then the probability of <em>k-1</em> tails in a row followed
by heads is given by the
\({\it Geometric}(p)\) distribution.
</p>
<a name="pdf_005fgeometric"></a><a name="Item_003a-distrib_002fdeffn_002fpdf_005fgeometric"></a><dl>
<dt><a name="index-pdf_005fgeometric"></a>Function: <strong>pdf_geometric</strong> <em>(<var>x</var>,<var>p</var>)</em></dt>
<dd><p>Returns the value at <var>x</var> of the probability function of a
\({\it Geometric}(p)\) random variable, with
<em>0 < p \leq 1</em>
</p>
<p>The pdf is
$$
f(x; p) = p(1-p)^x
$$</p>
<p>This is interpreted as the probability of <em>x</em> failures before the first success.
</p>
<p><code>load("distrib")</code> loads this function.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="cdf_005fgeometric"></a><a name="Item_003a-distrib_002fdeffn_002fcdf_005fgeometric"></a><dl>
<dt><a name="index-cdf_005fgeometric"></a>Function: <strong>cdf_geometric</strong> <em>(<var>x</var>,<var>p</var>)</em></dt>
<dd><p>Returns the value at <var>x</var> of the distribution function of a
\({\it Geometric}(p)\) random variable, with
<em>0 < p \leq 1</em>
</p>
<p>The cdf is
$$
1-(1-p)^{1 + \lfloor x \rfloor}
$$</p>
<p><code>load("distrib")</code> loads this function.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="quantile_005fgeometric"></a><a name="Item_003a-distrib_002fdeffn_002fquantile_005fgeometric"></a><dl>
<dt><a name="index-quantile_005fgeometric"></a>Function: <strong>quantile_geometric</strong> <em>(<var>q</var>,<var>p</var>)</em></dt>
<dd><p>Returns the <var>q</var>-quantile of a
\({\it Geometric}(p)\) random variable, with
<em>0 < p <= 1</em>;
in other words, this is the inverse of <code>cdf_geometric</code>.
Argument <var>q</var> must be an element of <em>[0,1]</em>.
</p>
<p>The probability from which the quantile is derived is defined as <em>p (1 - p)^x</em>.
This is interpreted as the probability of <em>x</em> failures before the first success.
</p>
<p><code>load("distrib")</code> loads this function.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="mean_005fgeometric"></a><a name="Item_003a-distrib_002fdeffn_002fmean_005fgeometric"></a><dl>
<dt><a name="index-mean_005fgeometric"></a>Function: <strong>mean_geometric</strong> <em>(<var>p</var>)</em></dt>
<dd><p>Returns the mean of a
\({\it Geometric}(p)\) random variable, with
<em>0 < p \leq 1</em>.
</p>
<p>The mean is
$$
E[X] = {1\over p} - 1
$$</p>
<p>The probability from which the mean is derived is defined as <em>p (1 - p)^x</em>.
This is interpreted as the probability of <em>x</em> failures before the first success.
</p>
<p><code>load("distrib")</code> loads this function.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="var_005fgeometric"></a><a name="Item_003a-distrib_002fdeffn_002fvar_005fgeometric"></a><dl>
<dt><a name="index-var_005fgeometric"></a>Function: <strong>var_geometric</strong> <em>(<var>p</var>)</em></dt>
<dd><p>Returns the variance of a
\({\it Geometric}(p)\) random variable, with
<em>0 < p \leq 1</em>.
</p>
<p>The variance is
$$
V[X] = {1-p\over p^2}
$$</p>
<p><code>load("distrib")</code> loads this function.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="std_005fgeometric"></a><a name="Item_003a-distrib_002fdeffn_002fstd_005fgeometric"></a><dl>
<dt><a name="index-std_005fgeometric"></a>Function: <strong>std_geometric</strong> <em>(<var>p</var>)</em></dt>
<dd><p>Returns the standard deviation of a
\({\it Geometric}(p)\) random variable, with
<em>0 < p \leq 1</em>.
</p>
$$
D[X] = {\sqrt{1-p} \over p}
$$
<p><code>load("distrib")</code> loads this function.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="skewness_005fgeometric"></a><a name="Item_003a-distrib_002fdeffn_002fskewness_005fgeometric"></a><dl>
<dt><a name="index-skewness_005fgeometric"></a>Function: <strong>skewness_geometric</strong> <em>(<var>p</var>)</em></dt>
<dd><p>Returns the skewness coefficient of a
\({\it Geometric}(p)\) random variable, with
<em>0 < p \leq 1</em>.
</p>
<p>The skewness coefficient is
$$
SK[X] = {2-p \over \sqrt{1-p}}
$$</p>
<p><code>load("distrib")</code> loads this function.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="kurtosis_005fgeometric"></a><a name="Item_003a-distrib_002fdeffn_002fkurtosis_005fgeometric"></a><dl>
<dt><a name="index-kurtosis_005fgeometric"></a>Function: <strong>kurtosis_geometric</strong> <em>(<var>p</var>)</em></dt>
<dd><p>Returns the kurtosis coefficient of a geometric random variable
\({\it Geometric}(p)\), with
<em>0 < p \leq 1</em>.
</p>
<p>The kurtosis coefficient is
$$
KU[X] = {p^2-6p+6 \over 1-p}
$$</p>
<p><code>load("distrib")</code> loads this function.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="random_005fgeometric"></a><a name="Item_003a-distrib_002fdeffn_002frandom_005fgeometric"></a><dl>
<dt><a name="index-random_005fgeometric"></a>Function: <strong>random_geometric</strong> <em>(<var>p</var>) <br> <tt>random_geometric</tt> (<var>p</var>,<var>n</var>)</em></dt>
<dd>
<p><code>random_geometric(<var>p</var>)</code> returns one random sample from a
\({\it Geometric}(p)\) distribution, with
<em>0 < p <= 1</em>.
</p>
<p><code>random_geometric(<var>p</var>, <var>n</var>)</code> returns a list of <var>n</var> random samples.
</p>
<p>The algorithm is based on simulation of Bernoulli trials.
</p>
<p>The probability from which the random sample is derived is defined as <em>p (1 - p)^x</em>.
This is interpreted as the probability of <em>x</em> failures before the first success.
</p>
<p><code>load("distrib")</code> loads this function.
</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_002fDiscrete-Uniform-Random-Variable"></a><hr>
<div class="header">
<p>
Next: <a href="maxima_245.html#Discrete-Uniform-Random-Variable" accesskey="n" rel="next">Discrete Uniform Random Variable</a>, Previous: <a href="maxima_243.html#Bernoulli-Random-Variable" accesskey="p" rel="previous">Bernoulli 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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