Standard Wald distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ standardWald (λ) is used to indicate that the random variable X has the standard Wald distribution with parameter λ. A standard Wald random variable X with parameter λ has probability density function

f(x) = \sqrt{\frac{λ}{2πx^{3}}}e^{− \frac{λ(x−1)^{2}}{2x}} x > 0,

for λ > 0. The probability density function for two different values of λ is illustrated below.

PSfrag replacements

λ = 0.4
λ = 3
x
f(x)

The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = ∫_{0}^{x}\sqrt{\frac{λ}{2πt^{3}}}e^{− \frac{λ(t−1)^{2}}{2t}} dt x > 0.

The survivor function, hazard function, inverse distribution, and cumulative hazard functions on the support of X are mathematically intractable. The moment generating function of X is

M(t) = E [e^{tX}] = e^{λ}( 1 − \sqrt{1 − \frac{2t}{λ}} ) t < \frac{λ}{2}.

The characteristic function of X is

ϕ(t) = E [e^{itX}] = e^{λ}( 1 − \sqrt{1 − \frac{2it}{λ}} ) t < \frac{λ}{2}.

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

E[X] = 1  V [X] = \frac{1}{λ}  E [( \frac{X − μ}{σ} ) ^{3}] = \frac{3}{\sqrt{λ}}  E [( \frac{X − μ}{σ} ) ^{4}] = 3 + \frac{15}{λ}.

APPL verification: The APPL statements

X:=[[x -¿ sqrt(lambda / (Pi * 2 * x ˆ 3)) * exp(-lambda(x - 1) ˆ 2 / (2 * x))],

[0,infinity], [”Continuous”, ”PDF”]]; CDF(X); Mean(X); Variance(X); Skewness(X); Kurtosis(X); MGF(X);

verify the cumulative distribution function and moment generating function but fail to yield the expected values.