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

f(x) = \frac{2xe^{−x^{2}/α}}{α} x > 0.

The Rayleigh distribution can be used to model the lifetime of an object or a service time. The probability density function with three different parameter settings is illustrated below.

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α = 0.5
α = 2
α = 8
x
f(x)

The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = 1 − e^{−x^{2}/α} x > 0.

The survivor function on the support of X is

S(x) = P(X ≥ x) = e^{−x^{2}/α} x > 0.

The hazard function on the support of X is

h(x) = \frac{f(x)}{S(x)} = \frac{2x}{α} x > 0.

The cumulative hazard function on the support of X is

H(x) = −lnS(x) = \frac{x^{2}}{α} x > 0.

The inverse distribution function of X is

F^{−1} (u) = \sqrt{−αln(1 − u)} 0 < u < 1.

The median of X is

\sqrt{αln(2)}.

The moment generating function of X is mathematically intractable.

The characteristic function of X is mathematically intractable.

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

E[X] = \frac{\sqrt{απ}}{2}  V [X] = \frac{α(4 − π)}{4}  E [( \frac{X − μ}{σ} ) ^{3}] = \frac{2\sqrt{π}(π − 3)}{(4 − π)^{3/2}}

E [( \frac{X − μ}{σ} ) ^{4}] = \frac{6π(4 − π) − 16}{(π − 4)^{2}} .

APPL verification: The APPL statements

assume(alpha ¿ 0); X:=[[x -¿ (2 * x / alpha) * exp(-x ˆ 2 / alpha)], [0,infinity],

[”Continuous”, ”PDF”]]; CDF(X); HF(X); CHF(X); Mean(X); Variance(X); Skewness(X); Kurtosis(X);

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