Introduction to Probability and Statistics for Engineers and Scientists

5.6Exponential Random Variables 175

1 – a a

0 za

FIGURE 5.9 P{Z>zα}=α.

The value ofzαcan, for anyα, be obtained from Table A1. For instance, since

1 − (1. 645)=.05

1 − (1. 96)=.025

1 − (2. 33)=.01

it follows that

z.05=1.645, z.025=1.96, z.01=2.33

Program 5.5b on the text disk can also be used to obtain the value ofzα.
Since
P{Z<zα}= 1 −α

it follows that 100(1−α) percent of the time a standard normal random variable will
be less thanzα. As a result, we callzαthe 100(1−α)percentileof the standard normal
distribution.

5.6Exponential Random Variables

A continuous random variable whose probability density function is given, for some
λ>0, by

f(x)=

{

λe−λx ifx≥ 0

0ifx< 0

is said to be anexponentialrandom variable (or, more simply, is said to be exponen-
tially distributed) with parameterλ. The cumulative distribution functionF(x)ofan
exponential random variable is given by

F(x)=P{X≤x}

=

∫x

0

λe−λydy

= 1 −e−λx, x≥ 0