Page 445 - A First Course In Stochastic Models

440 APPENDICES

not exist. For example, suppose that a n = 1 for n even and a n = 0 for n odd.

n
a
Then lim n→∞ a n does not exist, while lim n→∞ (1/n) k=1 k = 1/2. However, if
the ordinary limit exists then the Cesaro limit exists as well and is equal to the
ordinary limit.

APPENDIX B. USEFUL PROBABILITY DISTRIBUTIONS

This appendix discusses a number of important distributions which have been
found useful for describing random variables in inventory, reliability and queueing
applications. In particular, attention is paid to the practical problem of fitting a
tractable distribution to the first two moments of a positive random variable.

The exponential distribution
A positive random variable X is said to be exponentially distributed with parameter
λ > 0 when it has the probability density
−λt
f (t) = λe , t ≥ 0.
The corresponding probability distribution function F(t) is given by

F(t) = 1 − e −λt , t ≥ 0.

Its mean and squared coefficient of variation are given by
1 2
E(X) = and c = 1.
X
λ
The exponential distribution is of extreme importance in applied probability. The
main reason for this is its memoryless property and its intimate relation with the
Poisson process. The memoryless property states that

−λx
P {X > t + x | X > t} = e , x ≥ 0,
independently of t. In other words, imagining that X represents the lifetime of
an item, the residual life of the item has the same exponential distribution as the
original lifetime, regardless of how long the item has already been in use. The
memoryless property is in agreement with the constant failure rate property of the
exponential distribution.
The following well-known results for the exponential distribution are very use-
ful. If X 1 and X 2 are two independent random variables that are exponentially
distributed with respective means 1/λ 1 and 1/λ 2 , then, for any t ≥ 0,

P {min(X 1 , X 2 ) ≤ t} = 1 − e −(λ 1 +λ 2 )t and P {X 1 < X 2 } = λ 1 . (B.1)
λ 1 + λ 2