52.2.12 Logistic Random Variable

The logistic distribution is a continuous distribution where it’s cumulative distribution function is the logistic function.

Function: pdf_logistic (x,a,b) ¶

Returns the value at x of the density function of a \({\it Logistic}(a,b)\) random variable , with \(b>0\). To make use of this function, write first load("distrib").

\(a\) is the location parameter and \(b\) is the scale parameter.

The pdf is

\[f(x; a, b) = {e^{-(x-a)/b} \over b\left(1 + e^{-(x-a)/b}\right)^2} \]

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Function: cdf_logistic (x,a,b) ¶

Returns the value at x of the distribution function of a \({\it Logistic}(a,b)\) random variable , with \(b>0\). To make use of this function, write first load("distrib").

The cdf is

\[F(x; a, b) = {1\over 1+e^{-(x-a)/b}} \]

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Function: quantile_logistic (q,a,b) ¶

Returns the q-quantile of a \({\it Logistic}(a,b)\) random variable , with \(b>0\); in other words, this is the inverse of cdf_logistic. Argument q must be an element of \([0,1]\). To make use of this function, write first load("distrib").

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Function: mean_logistic (a,b) ¶

Returns the mean of a \({\it Logistic}(a,b)\) random variable , with \(b>0\). To make use of this function, write first load("distrib").

The mean is

\[E[X] = a \]

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Function: var_logistic (a,b) ¶

Returns the variance of a \({\it Logistic}(a,b)\) random variable , with \(b>0\). To make use of this function, write first load("distrib").

The variance is

\[V[X] = {\pi^2 b^2 \over 3} \]

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Function: std_logistic (a,b) ¶

Returns the standard deviation of a \({\it Logistic}(a,b)\) random variable , with \(b>0\). To make use of this function, write first load("distrib").

The standard deviation is

\[D[X] = {\pi b\over \sqrt{3}} \]

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Function: skewness_logistic (a,b) ¶

Returns the skewness coefficient of a \({\it Logistic}(a,b)\) random variable , with \(b>0\). To make use of this function, write first load("distrib").

The skewness coefficient is

\[SK[X] = 0 \]

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Function: kurtosis_logistic (a,b) ¶

Returns the kurtosis coefficient of a \({\it Logistic}(a,b)\) random variable , with \(b>0\). To make use of this function, write first load("distrib").

The kurtosis coefficient is

\[KU[X] = {6\over 5} \]

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Function: random_logistic (a,b)
    random_logistic (a,b,n) ¶

Returns a \({\it Logistic}(a,b)\) random variate, with \(b>0\). Calling random_logistic with a third argument n, a random sample of size n will be simulated.

The implemented algorithm is based on the general inverse method.

To make use of this function, write first load("distrib").

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