Laplace distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ Laplace(α_{1}, α_{2}) is used to indicate that the random variable X has the Laplace distribution with positive scale parameters α_{1} and α_{2}. A Laplace random variable X has probability density function

f(x) = \{ \begin{matrix}(1/(α_{1}+ α_{2}))e^{x/α_{1}} & x < 0 \\ (1/(α_{1}+ α_{2}))e^{−x/α_{2}} & x ≥ 0 \end{matrix}

for α_{1},α_{2} > 0. The Laplace distribution is an alternative to the normal distribution with heavier tails. The probability density function for three different parameter settings is illustrated below.

PSfrag replacements α_{1} = 1 α_{2} = 1 α_{1} = 2 α_{2} = 2

f(x)

The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = \{ \begin{matrix}(α_{1}/(α_{1}+ α_{2}))e^{x/α_{1}} & x < 0 \\ 1 − (α_{2}/(α_{1}+ α_{2}))e^{−x/α_{2}} & x ≥ 0. \end{matrix}

The survivor function on the support of X is

S(x) = P(X ≥ x) = \{ \begin{matrix}1 − (α_{1}/(α_{1}+ α_{2}))e^{x/α_{1}} & x < 0 \\ (α_{2}/(α_{1}+ α_{2}))e^{−x/α_{2}} & x ≥ 0. \end{matrix}

The hazard function on the support of X is

h(x) = \frac{f(x)}{S(x)} = \{ \begin{matrix}e^{x/α_{1}}/(α_{1}+ α_{2}+ α_{1}e^{x/α_{1}}) & x < 0 \\ 1/α_{2} & x ≥ 0. \end{matrix}

The inverse distribution function of X is

F^{−1} (u) = \{ \begin{matrix}−α_{1}(−ln(α_{1}+ α_{2}) + ln(α_{1}) − ln(u)) & 0 < u < α_{1}/(α_{1}+ α_{2}) \\ −α_{2}(ln(α_{1}+ α_{2}) − ln(α_{2}) + ln(1 − u)) & α_{1}/(α_{1}+ α_{2}) ≤ u < 1. \end{matrix}

The moment generating function and the characteristic function of X are mathematically intractable.

The population mean, variance, skewness, and kurtosis of X are

E[X] = α_{2}− α_{1} V [X] = α_{1}^{2}+ α_{2}^{2}

E [( \frac{X − μ}{σ} ) ^{3}] = −\frac{2(α_{1}^{3}− α_{2}^{3})}{(α_{1}^{2}+ α_{2}^{2})^{3/2}} E [( \frac{X − μ}{σ} ) ^{4}] = \frac{3(3α_{1}^{4}+ 2α_{2}^{2}α_{1}^{2}+ 3α_{2}^{4})}{(α_{1}^{2}+ α_{2}^{2})^{2}} .

APPL verification: The APPL statements

assume(alpha1 ¿ 0); assume(alpha2 ¿ 0); X := [[x -¿ exp(x / alpha1)/(alpha1 + alpha2),

x -¿ exp(-x / alpha2)/(alpha1 + alpha2) ], [-infinity, 0, infinity], [”Continuous”, ”PDF”]];

CDF(X); SF(X); HF(X); IDF(X); Mean(X); Variance(X); Skewness(X); Kurtosis(X);

verify the cumulative distribution function, survivor function, hazard function, population mean, variance, skewness, and kurtosis.