Applied Statistics and Probability for Engineers

130 CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

EXAMPLE 4-24 An alternative approach to computing the probability requested in Example 4-24 is to inte-

grate the probability density function of X. That is,

where Integration by parts can be used to verify the result obtained

previously.

An Erlang random variable can be thought of as the continuous analog of a negative

binomial random variable. A negative binomial random variable can be expressed as the sum

of rgeometric random variables. Similarly, an Erlang random variable can be represented as

the sum of rexponential random variables. Using this conclusion, we can obtain the follow-

ing plausible result. Sums of random variables are studied in Chapter 5.

r4 and 0.0001.

P 1 X40,000 2  



40,000

f 1 x 2 dx 



40,000

rxr^1 ex

1 r 12!

dx

0

0.0

0.4

0.8

1.2

1.6

2.0

2 4 6 8 10 12

x

f (x)

1

5

5

1

1

2

r λ

Figure 4-25 Erlang probability density functions

for selected values of rand .

If Xis an Erlang random variable with parameters and r,

E 1 X 2 r and ^2 V 1 X 2 r^2 (4-18)



4-10.2 Gamma Distribution

The Erlang distribution is a special case of the gamma distribution.If the parameter rof

an Erlang random variable is not an integer, but , the random variable has a gamma

distribution. However, in the Erlang density function, the parameter rappears as rfactorial.

r 0

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