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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