Introduction to Probability and Statistics for Engineers and Scientists

5.6Exponential Random Variables 179

=

∏n

i= 1

e−λix

=e−

∑n

i= 1 λix 

EXAMPLE 5.6c A series system is one that needs all of its components to function in
order for the system itself to be functional. For ann-component series system in which
the component lifetimes are independent exponential random variables with respective
parametersλ 1 ,λ 2 ,...,λn, what is the probability that the system survives for a timet?

SOLUTION Since the system life is equal to the minimal component life, it follows from
Proposition 5.6.1 that

P{system life exceedst}=e−

∑

iλit ■

Another useful property of exponential random variables is thatcXis exponential with
parameterλ/cwhenXis exponential with parameterλ, andc>0. This follows since

P{cX≤x}=P{X≤x/c}

= 1 −e−λx/c

The parameterλis called therateof the exponential distribution.

*5.6.1 The Poisson Process

Suppose that “events” are occurring at random time points, and letN(t) denote the number
of events that occurs in the time interval [0,t]. These events are said to constitute aPoisson
process having rateλ,λ>0, if

(a) N(0)= 0

(b)The numbers of events that occur in disjoint time intervals are independent.

(c) The distribution of the number of events that occur in a given interval depends

only on the length of the interval and not on its location.

(d) lim

h→ 0

P{N(h)= 1 }

h

=λ

(e) lim

h→ 0

P{N(h)≥ 2 }

h

= 0

Thus, Condition (a) states that the process begins at time 0. Condition (b), theinde-
pendent incrementassumption, states for instance that the number of events by timet
[that is,N(t)] is independent of the number of events that occurs betweentandt+s
[that is,N(t+s)−N(t)]. Condition (c), thestationary incrementassumption, states that
probability distribution ofN(t+s)−N(t) is the same for all values oft. Conditions (d)

* Optional section.