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<a name="Logistic-Random-Variable"></a>
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<p>
Next: <a href="maxima_233.html#Pareto-Random-Variable" accesskey="n" rel="next">Pareto Random Variable</a>, Previous: <a href="maxima_231.html#Continuous-Uniform-Random-Variable" accesskey="p" rel="previous">Continuous Uniform 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="Logistic-Random-Variable-1"></a>
<h4 class="subsection">52.2.12 Logistic Random Variable</h4>
<p>The <em>logistic</em> distribution is a continuous distribution where
it’s cumulative distribution function is the logistic function.
</p>
<a name="pdf_005flogistic"></a><a name="Item_003a-distrib_002fdeffn_002fpdf_005flogistic"></a><dl>
<dt><a name="index-pdf_005flogistic"></a>Function: <strong>pdf_logistic</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
\({\it Logistic}(a,b)\) random variable , with <em>b>0</em>. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p><em>a</em> is the location parameter and <em>b</em> is the scale
parameter.
</p>
<p>The pdf is
$$
f(x; a, b) = {e^{-(x-a)/b} \over b\left(1 + e^{-(x-a)/b}\right)^2}
$$</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="cdf_005flogistic"></a><a name="Item_003a-distrib_002fdeffn_002fcdf_005flogistic"></a><dl>
<dt><a name="index-cdf_005flogistic"></a>Function: <strong>cdf_logistic</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
\({\it Logistic}(a,b)\) random variable , with <em>b>0</em>. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The cdf is
$$
F(x; a, b) = {1\over 1+e^{-(x-a)/b}}
$$</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="quantile_005flogistic"></a><a name="Item_003a-distrib_002fdeffn_002fquantile_005flogistic"></a><dl>
<dt><a name="index-quantile_005flogistic"></a>Function: <strong>quantile_logistic</strong> <em>(<var>q</var>,<var>a</var>,<var>b</var>)</em></dt>
<dd><p>Returns the <var>q</var>-quantile of a
\({\it Logistic}(a,b)\) random variable , with <em>b>0</em>; in other words, this is the inverse of <code>cdf_logistic</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_005flogistic"></a><a name="Item_003a-distrib_002fdeffn_002fmean_005flogistic"></a><dl>
<dt><a name="index-mean_005flogistic"></a>Function: <strong>mean_logistic</strong> <em>(<var>a</var>,<var>b</var>)</em></dt>
<dd><p>Returns the mean of a
\({\it Logistic}(a,b)\) random variable , with <em>b>0</em>. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The mean is
$$
E[X] = a
$$</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
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</dd></dl>
<a name="var_005flogistic"></a><a name="Item_003a-distrib_002fdeffn_002fvar_005flogistic"></a><dl>
<dt><a name="index-var_005flogistic"></a>Function: <strong>var_logistic</strong> <em>(<var>a</var>,<var>b</var>)</em></dt>
<dd><p>Returns the variance of a
\({\it Logistic}(a,b)\) random variable , with <em>b>0</em>. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The variance is
$$
V[X] = {\pi^2 b^2 \over 3}
$$</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
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</dd></dl>
<a name="std_005flogistic"></a><a name="Item_003a-distrib_002fdeffn_002fstd_005flogistic"></a><dl>
<dt><a name="index-std_005flogistic"></a>Function: <strong>std_logistic</strong> <em>(<var>a</var>,<var>b</var>)</em></dt>
<dd><p>Returns the standard deviation of a
\({\it Logistic}(a,b)\) random variable , with <em>b>0</em>. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The standard deviation is
$$
D[X] = {\pi b\over \sqrt{3}}
$$</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="skewness_005flogistic"></a><a name="Item_003a-distrib_002fdeffn_002fskewness_005flogistic"></a><dl>
<dt><a name="index-skewness_005flogistic"></a>Function: <strong>skewness_logistic</strong> <em>(<var>a</var>,<var>b</var>)</em></dt>
<dd><p>Returns the skewness coefficient of a
\({\it Logistic}(a,b)\) random variable , with <em>b>0</em>. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The skewness coefficient is
$$
SK[X] = 0
$$</p>
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Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="kurtosis_005flogistic"></a><a name="Item_003a-distrib_002fdeffn_002fkurtosis_005flogistic"></a><dl>
<dt><a name="index-kurtosis_005flogistic"></a>Function: <strong>kurtosis_logistic</strong> <em>(<var>a</var>,<var>b</var>)</em></dt>
<dd><p>Returns the kurtosis coefficient of a
\({\it Logistic}(a,b)\) random variable , with <em>b>0</em>. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The kurtosis coefficient is
$$
KU[X] = {6\over 5}
$$</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-distrib">Package distrib</a>
·</div>
</dd></dl>
<a name="random_005flogistic"></a><a name="Item_003a-distrib_002fdeffn_002frandom_005flogistic"></a><dl>
<dt><a name="index-random_005flogistic"></a>Function: <strong>random_logistic</strong> <em>(<var>a</var>,<var>b</var>) <br> <tt>random_logistic</tt> (<var>a</var>,<var>b</var>,<var>n</var>)</em></dt>
<dd>
<p>Returns a
\({\it Logistic}(a,b)\) random variate, with <em>b>0</em>. Calling <code>random_logistic</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("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_002fPareto-Random-Variable"></a><hr>
<div class="header">
<p>
Next: <a href="maxima_233.html#Pareto-Random-Variable" accesskey="n" rel="next">Pareto Random Variable</a>, Previous: <a href="maxima_231.html#Continuous-Uniform-Random-Variable" accesskey="p" rel="previous">Continuous Uniform 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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