Page 448 - A First Course In Stochastic Models

B. USEFUL PROBABILITY DISTRIBUTIONS 443

random variables each having an exponential distribution with the same mean 1/λ;
see also Appendix E. The Erlang probability distribution function can be numeri-
cally evaluated without using a general code for the incomplete gamma integral. For
j
fixed t, the Poisson probabilities p j (t) = e −λt (λt) /j! can be recursively calculated
from p 0 (t) = e −λt and p j (t) = (λt/j)p j−1 (t) for j = 1, 2, . . . . However, expo-
nent underflow may occur in the calculation of p 0 (t) when λt is very large. There

is a simple trick to avoid the exponent underflow. Define q j (t) = ln p j (t) . The
recursion scheme q 0 (t) = −λt and q j (t) = ln(λt/j) + q j−1 (t) for j ≥ 1 offers no
numerical difficulties at all. Any desired p j (t) is calculated from p j (t) = exp[q j (t)]
if q j (t) ≥ −100 (say) and p j (t) = 0 otherwise. The trick of working with loga-
rithms is one of the most useful tricks to avoid underflow in numerical analysis.
Logarithms enable us to reduce the manipulation with extremely large (small)
numbers to the manipulation with moderately sized numbers.

The lognormal distribution
A positive random variable X is said to be lognormally distributed when it has the
probability density
1 1 2 2
f (t) = √ exp − [ln(t) − λ] /α , t > 0,
αt 2π 2
where the shape parameter α is positive and the scale parameter λ may assume
each real value. The probability density function F(t) equals

ln(t) − λ
F(t) = , t > 0,
α
√ x
2
where (x) = (1/ 2π) exp(−u /2) du is the standard normal probability
−∞
distribution function. The mean and the squared coefficient of variation of the
lognormal distribution are given by

1 2 2 2
E(X) = exp λ + α and c = exp(α ) − 1.
X
2
Thus a unique lognormal distribution can be fitted to each positive random variable
with given first two moments. The lognormal density is always unimodal with a
2
maximum at t = exp(λ − α ). The failure rate function first increases and next
decreases to zero as t → ∞ and thus the failure rate is only decreasing in the
long-life range.

The Weibull distribution
A positive random variable X is said to be Weibull distributed when it has the
probability density
α
f (t) = αλ(λt) α−1 exp[−(λt) ], t > 0,