Hypoexponential distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ hypoexponential(α⃗) is used to indicate that the random variable X has the hypoexponential distribution with positive vector parameter α⃗ such that α_{i} ≠ α_{j} for i ≠ j. If T_{i} ∼ exponential(α_{i}) for i = 1,2,...,n, and T_{1}, T_{2}, ..., T_{n} are mutually independent random variables, then T = T_{1}+T_{2}+··· +T_{n} has the hypoexponential distribution. A hypoexponential random variable X has probability density function

f(x) = ∑_{i=1}^{n} (1/α_{i})e^{−x/α_{i}} ( ∏_{j =1,j ≠i}^{n} \frac{α_{i}}{α_{i}− α_{j}} )

for x > 0. The probability density function for α⃗ = (0.25, 0.5, 1) is illustrated below.

PSfrag replacements

f(x)

x

For the case where n = 2, the cumulative distribution function on the support of X is

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

The general cumulative distribution, survivor, hazard, cumulative hazard, moment generating, and characteristic functions on the support of X are mathematically intractable.

The population mean, variance, and skewness of X are

E[X] = ∑_{i=1}^{n} α_{i}  V [X] = ∑_{i=1}^{n} α_{i}^{2}  E [( \frac{X − μ}{σ} ) ^{3}] = \frac{2 ∑_{i=1}^{n} α_{i}^{3}}{( ∑_{i=1}^{n} α_{i}^{2}) ^{3/2}}.

APPL verification: The APPL statements

X:= HypoExponentialRV([1 / alpha1, 1 / alpha2, 1 / alpha3]); Mean(X); Variance(X); Skewness(X);

verify the population mean, variance, and skewness for the special case of n = 3.