Page 443 - A First Course In Stochastic Models

438 APPENDICES

Since this quantity is dimensionless, it is a very useful measure for the variability of
the random variable X. Usually one works with the squared coefficient of variation
c 2 rather than with c X . For example, the deterministic distribution has c 2 = 0,
X X
the exponential distribution has c 2 = 1 and the Erlang distribution with shape
X
2
parameter k has the intermediate value c = 1/k.
X
Failure rate function
Let X be a positive random variable with a probability distribution function F(t)
and a continuous probability density f (t). For example, the random variable X
represents the lifetime of some item. The failure, or hazard, rate function of the
random variable X is defined by
f (t)
r(t) =
1 − F(t)
for those values of t with F(t) < 1. The failure rate has a useful probabilistic
interpretation. Think of the random variable X as the lifetime of an item. The
probability that an item of age t will fail in the next t time units is given by

P {t < X ≤ t + t}
P {t < X ≤ t + t | X > t} =
P {X > t}
f (t) t
= + o( t) as t → 0.
1 − F(t)
Hence r(t) t gives approximately the probability that an item of age t will fail in
the next t time units when t is small. Hence the name ‘failure rate’. Noting that
−r(t) is the derivative of the function ln[1 − F(t)], it follows that the failure rate
function r(t) determines uniquely the corresponding lifetime distribution function
F(t) by
t
1 − F(t) = exp − r(x) dx , t ≥ 0.
0
As a consequence, the case of a constant failure rate r(x) = λ for all x corresponds
to the exponential distribution function F(x) = 1−e −λx , x ≥ 0. In other words, an
item in use is as good as new when its lifetime is exponentially distributed. Other
important cases are the case of an increasing failure rate (the older, the worse) and
the case of a decreasing failure rate (the older, the better). A random variable with
an increasing (decreasing) failure rate can be shown to have the property that its
coefficient of variation is smaller (larger) than 1. The failure rate is a concept that
enables us to discriminate between distributions on physical considerations.

Convergence theorems
To conclude this appendix, we state a number of basic convergence theorems that
will be used in this book. These theorems can be found in any textbook on real
analysis, e.g. Rudin (1964).