52.2.16 Laplace Random Variable

The Laplace distribution is a continuous probability distribution that is sometimes called the double exponential distribution because it can be thought of as two exponential distributions spliced back to back.

Function: pdf_laplace (x,a,b) ¶

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

Here, \(a\) is the location parameter (or mean), and \(b\) is the scale parameter, related to the variance.

The pdf is

\[f(x; a, b) = {1\over 2b}\exp\left(-{|x-a|\over b}\right) \]

Categories: Package distrib ·

Function: cdf_laplace (x,a,b) ¶

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

The cdf is

\[F(x; a, b) = \cases{ \displaystyle{1\over 2} \exp\left({x-a\over b}\right) & for $x < a$\cr \cr 1-\displaystyle{1\over 2} \exp\left({x-a\over b}\right) & for $x \ge a$ } \]

Categories: Package distrib ·

Function: quantile_laplace (q,a,b) ¶

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

Categories: Package distrib ·

Function: mean_laplace (a,b) ¶

Returns the mean of a \({\it Laplace}(a,b)\) 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_laplace (a,b) ¶

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

The variance is

\[V[X] = 2b^2 \]

Categories: Package distrib ·

Function: std_laplace (a,b) ¶

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

The standard deviation is

\[D[X] = \sqrt{2} b \]

Categories: Package distrib ·

Function: skewness_laplace (a,b) ¶

Returns the skewness coefficient of a \({\it Laplace}(a,b)\) 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_laplace (a,b) ¶

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

The kurtosis coefficient is

\[KU[X] = 3 \]

Categories: Package distrib ·

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

Returns a \({\it Laplace}(a,b)\) random variate, with \(b>0\). Calling random_laplace 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 ·