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<a name="Pareto-Random-Variable"></a>
<div class="header">
<p>
Next: <a href="maxima_234.html#Weibull-Random-Variable" accesskey="n" rel="next">Weibull Random Variable</a>, Previous: <a href="maxima_232.html#Logistic-Random-Variable" accesskey="p" rel="previous">Logistic 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> &nbsp; [<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="Pareto-Random-Variable-1"></a>
<h4 class="subsection">52.2.13 Pareto Random Variable</h4>

<a name="pdf_005fpareto"></a><a name="Item_003a-distrib_002fdeffn_002fpdf_005fpareto"></a><dl>
<dt><a name="index-pdf_005fpareto"></a>Function: <strong>pdf_pareto</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 
\({\it Pareto}(a,b)\) random variable, with <em>a,b&gt;0</em>. To make use of this function, write first <code>load(&quot;distrib&quot;)</code>.
</p>
<p>The pdf is
$$
f(x; a, b) = 
\cases{
\displaystyle{a b^a \over x^{a+1}} & for $x \ge b$ \cr
\cr
0 & for $x < b$
}
$$</p>

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
&middot;</div>
</dd></dl>


<a name="cdf_005fpareto"></a><a name="Item_003a-distrib_002fdeffn_002fcdf_005fpareto"></a><dl>
<dt><a name="index-cdf_005fpareto"></a>Function: <strong>cdf_pareto</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 
\({\it Pareto}(a,b)\) random variable, with <em>a,b&gt;0</em>. To make use of this function, write first <code>load(&quot;distrib&quot;)</code>.
</p>
<p>The cdf is
$$
F(x; a, b) =
\cases{
1-\left(\displaystyle{b\over x}\right)^a & for $x \ge b$\cr
0 & for $x < b$
}
$$</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
&middot;</div>
</dd></dl>


<a name="quantile_005fpareto"></a><a name="Item_003a-distrib_002fdeffn_002fquantile_005fpareto"></a><dl>
<dt><a name="index-quantile_005fpareto"></a>Function: <strong>quantile_pareto</strong> <em>(<var>q</var>,<var>a</var>,<var>b</var>)</em></dt>
<dd><p>Returns the <var>q</var>-quantile of a 
\({\it Pareto}(a,b)\) random variable, with <em>a,b&gt;0</em>; in other words, this is the inverse of <code>cdf_pareto</code>. Argument <var>q</var> must be an element of <em>[0,1]</em>. To make use of this function, write first <code>load(&quot;distrib&quot;)</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
&middot;</div>
</dd></dl>


<a name="mean_005fpareto"></a><a name="Item_003a-distrib_002fdeffn_002fmean_005fpareto"></a><dl>
<dt><a name="index-mean_005fpareto"></a>Function: <strong>mean_pareto</strong> <em>(<var>a</var>,<var>b</var>)</em></dt>
<dd><p>Returns the mean of a 
\({\it Pareto}(a,b)\) random variable, with <em>a&gt;1,b&gt;0</em>. To make use of this function, write first <code>load(&quot;distrib&quot;)</code>.
</p>
<p>The mean is
$$
E[X] = {ab\over a-1}
$$</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="var_005fpareto"></a><a name="Item_003a-distrib_002fdeffn_002fvar_005fpareto"></a><dl>
<dt><a name="index-var_005fpareto"></a>Function: <strong>var_pareto</strong> <em>(<var>a</var>,<var>b</var>)</em></dt>
<dd><p>Returns the variance of a 
\({\it Pareto}(a,b)\) random variable, with <em>a&gt;2,b&gt;0</em>. To make use of this function, write first <code>load(&quot;distrib&quot;)</code>.
</p>
<p>The variance is
$$
V[X] = {ab^2\over (a-2)(a-1)^2}
$$</p>

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
&middot;</div>
</dd></dl>

<a name="std_005fpareto"></a><a name="Item_003a-distrib_002fdeffn_002fstd_005fpareto"></a><dl>
<dt><a name="index-std_005fpareto"></a>Function: <strong>std_pareto</strong> <em>(<var>a</var>,<var>b</var>)</em></dt>
<dd><p>Returns the standard deviation of a 
\({\it Pareto}(a,b)\) random variable, with <em>a&gt;2,b&gt;0</em>. To make use of this function, write first <code>load(&quot;distrib&quot;)</code>.
</p>
<p>The standard deviation is
$$
D[X] = {b\over a-1} \sqrt{a\over a-2}
$$</p>

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Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
&middot;</div>
</dd></dl>



<a name="skewness_005fpareto"></a><a name="Item_003a-distrib_002fdeffn_002fskewness_005fpareto"></a><dl>
<dt><a name="index-skewness_005fpareto"></a>Function: <strong>skewness_pareto</strong> <em>(<var>a</var>,<var>b</var>)</em></dt>
<dd><p>Returns the skewness coefficient of a 
\({\it Pareto}(a,b)\) random variable, with <em>a&gt;3,b&gt;0</em>. To make use of this function, write first <code>load(&quot;distrib&quot;)</code>.
</p>
<p>The skewness coefficient is
$$
SK[X] = {2(a+1)\over a-3} \sqrt{a-2\over a}
$$</p>

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
&middot;</div>
</dd></dl>


<a name="kurtosis_005fpareto"></a><a name="Item_003a-distrib_002fdeffn_002fkurtosis_005fpareto"></a><dl>
<dt><a name="index-kurtosis_005fpareto"></a>Function: <strong>kurtosis_pareto</strong> <em>(<var>a</var>,<var>b</var>)</em></dt>
<dd><p>Returns the kurtosis coefficient of a 
\({\it Pareto}(a,b)\) random variable, with <em>a&gt;4,b&gt;0</em>. To make use of this function, write first <code>load(&quot;distrib&quot;)</code>.
</p>
<p>The kurtosis coefficient is
$$
KU[X] = {6\left(a^3+a^2-6*a-2\right) \over a(a-3)(a-4)} - 3
$$</p>

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
&middot;</div>
</dd></dl>


<a name="random_005fpareto"></a><a name="Item_003a-distrib_002fdeffn_002frandom_005fpareto"></a><dl>
<dt><a name="index-random_005fpareto"></a>Function: <strong>random_pareto</strong> <em>(<var>a</var>,<var>b</var>) <br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>random_pareto</tt> (<var>a</var>,<var>b</var>,<var>n</var>)</em></dt>
<dd>
<p>Returns a 
\({\it Pareto}(a,b)\) random variate, with <em>a&gt;0,b&gt;0</em>. Calling <code>random_pareto</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 based on the general inverse method.
</p>
<p>To make use of this function, write first <code>load(&quot;distrib&quot;)</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
&middot;<a href="maxima_424.html#Category_003a-Random-numbers">Random numbers</a>
&middot;</div>
</dd></dl>


<a name="Item_003a-distrib_002fnode_002fWeibull-Random-Variable"></a><hr>
<div class="header">
<p>
Next: <a href="maxima_234.html#Weibull-Random-Variable" accesskey="n" rel="next">Weibull Random Variable</a>, Previous: <a href="maxima_232.html#Logistic-Random-Variable" accesskey="p" rel="previous">Logistic 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> &nbsp; [<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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