Hyperexponential distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ hyperexponential (α⃗, p⃗) is used to indicate that the random variable X has the hyperexponential distribution with parameters α⃗ and p⃗. A hyperexponential random variable X with parameters α⃗ and p⃗ has probability density function

f(x) = ∑_{i=1}^{n} \frac{p_{i}}{α_{i}} e^{−x/α_{i}} x > 0

for all α_{i},p_{i} > 0 such that ∑ _{i=1}^{n} p_{i} = 1. The probability density function for α⃗ = (0.25, 0.5, 1) and p⃗ = (0.5, 0.25, 0.25) is illustrated below.

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f(x)

x

The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = 1 − ∑_{i=1}^{n} p_{i}e^{−x/α_{i}} x > 0.

The survivor function on the support of X is

S(x) = P(X ≤ x) = ∑_{i=1}^{n} p_{i}e^{−x/α_{i}} x > 0.

The hazard function on the support of X is

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

The inverse distribution function and characteristic function are both mathematically intractable.

The moment generating function over the support of X is

M(t) = E[e^{tX}] = ∑_{i=1}^{n} \frac{p_{i}}{1 − α_{i}t} |t| < \frac{1}{min_{j} α_{j}} .

The population mean of X is

E[X] = ∑_{i=1}^{n} p_{i}α_{i}.

The population skewness and kurtosis of X are all mathematically intractable.

APPL verification: The APPL statements

X := HyperExponentialRV([0.1, 0.3, 0.6], [alpha1, alpha2, alpha3]); CDF(X); SF(X); Mean(X); Variance(X); Skewness(X); Kurtosis(X); MGF(X);

verify the population mean, variance, skewness, kurtosis, and moment generating function for the special case of n = 3 and p⃗ = (0.1, 0.3, 0.6).