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 $\mathit{L}\mathit{o}\mathit{g}\mathit{i}\mathit{s}\mathit{t}\mathit{i}\mathit{c}(a,b)$ 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

$$f(x;a,b)=\frac{e^{-(x-a)/b}}{b{(1+e^{-(x-a)/b})}^2}$$

Categories: Package distrib ·

Function: cdf_logistic (x,a,b) ¶

Returns the value at x of the distribution function of a $\mathit{L}\mathit{o}\mathit{g}\mathit{i}\mathit{s}\mathit{t}\mathit{i}\mathit{c}(a,b)$ random variable , with $b>0$. To make use of this function, write first load("distrib").

The cdf is

$$F(x;a,b)=\frac{1}{1+e^{-(x-a)/b}}$$

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Function: quantile_logistic (q,a,b) ¶

Returns the q-quantile of a $\mathit{L}\mathit{o}\mathit{g}\mathit{i}\mathit{s}\mathit{t}\mathit{i}\mathit{c}(a,b)$ 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").

Categories: Package distrib ·

Function: mean_logistic (a,b) ¶

Returns the mean of a $\mathit{L}\mathit{o}\mathit{g}\mathit{i}\mathit{s}\mathit{t}\mathit{i}\mathit{c}(a,b)$ random variable , with $b>0$. To make use of this function, write first load("distrib").

The mean is

$$E[X]=a$$

Categories: Package distrib ·

Function: var_logistic (a,b) ¶

Returns the variance of a $\mathit{L}\mathit{o}\mathit{g}\mathit{i}\mathit{s}\mathit{t}\mathit{i}\mathit{c}(a,b)$ random variable , with $b>0$. To make use of this function, write first load("distrib").

The variance is

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

Categories: Package distrib ·

Function: std_logistic (a,b) ¶

Returns the standard deviation of a $\mathit{L}\mathit{o}\mathit{g}\mathit{i}\mathit{s}\mathit{t}\mathit{i}\mathit{c}(a,b)$ random variable , with $b>0$. To make use of this function, write first load("distrib").

The standard deviation is

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

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Function: skewness_logistic (a,b) ¶

Returns the skewness coefficient of a $\mathit{L}\mathit{o}\mathit{g}\mathit{i}\mathit{s}\mathit{t}\mathit{i}\mathit{c}(a,b)$ random variable , with $b>0$. To make use of this function, write first load("distrib").

The skewness coefficient is

$$SK[X]=0$$

Categories: Package distrib ·

Function: kurtosis_logistic (a,b) ¶

Returns the kurtosis coefficient of a $\mathit{L}\mathit{o}\mathit{g}\mathit{i}\mathit{s}\mathit{t}\mathit{i}\mathit{c}(a,b)$ random variable , with $b>0$. To make use of this function, write first load("distrib").

The kurtosis coefficient is

$$KU[X]=\frac{6}{5}$$

Categories: Package distrib ·

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

Returns a $\mathit{L}\mathit{o}\mathit{g}\mathit{i}\mathit{s}\mathit{t}\mathit{i}\mathit{c}(a,b)$ 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").

Categories: Package distrib · Random numbers ·