Applied Statistics and Probability for Engineers

124 CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

Also, the cumulative distribution function can be used to obtain the same result as follows:

An identical answer is obtained by expressing the mean number of log-ons as 0.417 log-

ons per minute and computing the probability that the time until the next log-on exceeds 6

minutes. Try it.

What is the probability that the time until the next log-on is between 2 and 3 minutes?

Upon converting all units to hours,

An alternative solution is

Determine the interval of time such that the probability that no log-on occurs in the inter-

val is 0.90. The question asks for the length of time xsuch that. Now,

Take the (natural) log of both sides to obtain. Therefore,

x0.00421 hour0.25 minute

 25 xln 1 0.90 2 0.1054

P 1 Xx 2 e^25 x0.90

P 1 Xx 2 0.90

P 1 0.033X0.05 2 F 1 0.05 2 F 1 0.033 2 0.152

P 1 0.033X0.05 2  

0.05

0.033

25 e^25 x dxe^25 x `

0.05

0.033

0.152

P 1 X0.1 2  1 F 1 0.1 2 e^251 0.1^2

0

0.0

0.4

0.8

1.2

1.6

2.0

2 4 6 8 10 12

x

f(x)

2

0.5

0.1

λ

Figure 4-22 Probability density function of expo-

nential random variables for selected values of .

0.1 x

f(x)

Figure 4-23 Probability for the expo-

nential distribution in Example 4-21.

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