Erlang Distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ Erlang(n,α) is used to indicate that the random variable X has the Erlang distribution with positive shape parameter n and positive scale parameter α. An Erlang random variable X with rate α and n stages has probability density function

f(x) = \frac{x^{n−1} e^{−x/α}}{α^{n}(n − 1)!} x > 0.

The Erlang distribution can be used to model the time to complete n operations in series, where each operation requires an exponential period of time to complete. The probability density function for various values of n and α is illustrated below.

PSfrag replacements n = 1, α = 1

n = 3, α = 1
n = 1, α = 4
x
f(x)

The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = 1 − ∑_{i=0}^{n−1} \frac{e^{−x/α} x^{i}}{α^{i}i!}

The survivor function on the support of X is

S(x) = P(X ≥ x) = ∑_{i=0}^{n−1} \frac{e^{−x/α} x^{i}}{α^{i}i!} x > 0.

The hazard function on the support of X is

h(x) = \frac{f(x)}{S(x)} = \frac{x^{n−1}}{α^{n}(n − 1)!∑ _{i=0}^{n−1} x^{i}/(α^{i}i!)} x > 0.

The cumulative hazard function of X is H(x) = −lnS(x) is not mathematically tractable. The inverse distribution function of X is also intractable, although random variates can be generated via

X ← −αln( ∏_{i=1}^{n} U_{i}) ,

where U_{1},U_{2},...,U_{n} are independent and identically distributed U(0,1) random variates. There is no closed-form expression for the median. The moment generating function of X is

M(t) = E [e^{tX}] = (1 − αt) ^{−n} t ≤ α.

The characteristic function of X is

ϕ(t) = E [e^{itX}] = (1 − iαt)^{−n} t < α.

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

E[X] = nα  V [X] = nα^{2}  E [( \frac{X − μ}{σ} ) ^{3}] = \frac{2}{\sqrt{n}}  E [( \frac{X − μ}{σ} ) ^{4}] = 3+ \frac{6}{n}.

APPL verification: The APPL statements

assume(n, posint); assume(alpha ¿ 0); X:=[[x -¿ x ˆ (n - 1) * exp(-x / alpha) / (alpha ˆ n * (n-1)!)], [0, infinity],

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

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