The Weibull Distribution

Closely related to the exponential distribution is the Weibull distribution, whose probability density is given by

Weibull distribution f (x)=

α βx^β−1e−αx^β for x> 0, α > 0, β > 0

0 elsewhere

Sec 5.9 The Weibull Distribution 159

The Weibull distribution has cumulative distribution function F ( x ) = 1 − e−αx^β x > 0 which is obtained from

F ( x ) =

 _x

0 α β w^β−1e−αw^βdw by making the change of variable y = w^β. Then

F ( x ) =

 _xβ

0 α e^{− αy}dy = 1 − e−αx^β. If X has the Weibull distribution and Y = X^β, then

P ( X^β ≤ y ) = P ( X ≤ y^1/β) = 1 − e−α(y^1/β)^β = 1 − e^−αy which is the cumulative distribution of the exponential distribution. That is, when X has the Weibull distribution then Y = X^β has an exponential distribution. The graphs of several Weibull distributions withα = 1 and β = ¹₂, 1, and 2 are shown in Figure 5.23.

Figure 5.23

Graphs of Weibull densities withα = 1 and β = 1

2, 1, and 2

b 5 0.5

b 5 1 b 5 2

0 x

f(x)

The mean of the Weibull distribution having the parametersα and β may be obtained by evaluating the integral

μ =

 _∞ 0

x· αβx^β−1e^−αxβ dx Making the change of variable u= αx^β, we get

μ = α^−1/β

 _∞ 0

u^1/βe^−udu Recognizing the integral as



1+ 1 β

where the gamma function is defined on page 155, we obtain the mean of the Weibull distribution.

Mean of Weibull

Using a similar method to determine firstμ^₂, the reader will be asked to show in Exercise 5.70 that the variance of this distribution is given by

Variance of Weibull

EXAMPLE 19 Probability calculations using a Weibull distribution

Suppose that the lifetime of a certain kind of an emergency backup battery (in hours) is a random variable X having the Weibull distribution withα = 0.1 and β = 0.5.

Find

(a) the mean lifetime of these batteries;

(b) the probability that such a battery will last more than 300 hours.

Solution (a) Substitution into the formula for the mean yields μ = (0.1)⁻²(3) = 200 hours (b) Performing the necessary integration, we get

 _∞ 300

(0.05)x^−0.5e^−0.1x0.5dx= e^{−0.1(300)0.5}

= 0.177 ^j

Exercises

5.45 Find the distribution function of a random variable having a uniform distribution on (0, 1).

5.46 In a manufacturing process, the error made in deter-mining the composition of an alloy is a random vari-able having the uniform density withα = −0.075 and β = 0.010. What are the probabilities that such an error will be

(a) between 0.050 and 0.001?

(b) between 0.001 and 0.008?

5.47 From experience Mr. Harris has found that the low bid on a construction job can be regarded as a random vari-able having the uniform density

f (x)=

where C is his own estimate of the cost of the job. What percentage should Mr. Harris add to his cost estimate when submitting bids to maximize his expected profit?

5.48 Verify the expression given on page 154 for the mean of the log-normal distribution.

5.49 With reference to the Example 12, find the probability that Io/Iiwill take on a value between 7.0 and 7.5.

5.50 If a random variable has the log-normal distribution withα = −3 and β = 3, find its mean and its stan-dard deviation.

5.51 With reference to the preceding exercise, find the prob-abilities that the random variable will take on a value (a) less than 8.0;

(b) between 4.5 and 6.5.

5.52 If a random variable has the gamma distribution with α = 2 and β = 3, find the mean and the standard de-viation of this distribution.

5.53 With reference to Exercise 5.52, find the probabil-ity that the random variable will take on a value less than 5.

5.54 At a construction site, the daily requirement of gneiss (in metric tons) is a random variable having a gamma

Sec 5.10 Joint Distributions—Discrete and Continuous 161

distribution withα = 2 and β = 5. If their supplier’s daily supply capacity is 25 metric tons, what is the probability that this capacity will be inadequate on any given day?

5.55 With reference to the Example 14, suppose the expert opinion is in error. Calculate the probability that the supports will survive if

(a) μ = 3.0 and σ²= 0.09;

(b) μ = 4.0 and σ²= 0.25.

5.56 Verify the expression for the variance of the gamma distribution given on page 156.

5.57 Show that whenα > 1, the graph of the gamma den-sity has a relative maximum at x= β(α − 1). What happens when 0< α < 1 and when α = 1?

5.58 The server of a multinational corporate network can run for an amount of time without having to be re-booted and this amount of time is a random variable having the exponential distributionβ = 30 days. Find the probabilities that such a server will

(a) have to be rebooted in less than 10 days;

(b) not have to be rebooted in at least 45 days.

5.59 With reference to Exercise 4.95, find the percent of the time that the interval between breakdowns of the computer will be

(a) less than 1 week;

(b) at least 5 weeks.

5.60 With reference to Exercise 4.58, find the probabilities that the time between successive requests for consult-ing will be

(a) less than 0.5 week;

(b) more than 3 weeks.

5.61 Given a Poisson process with on the averageα arrivals per unit time, find the probability that there will be no arrivals during a time interval of length t, namely, the probability that the waiting times between successive arrivals will be at least of length t.

5.62 Use the result of Exercise 5.61 to find an expression for the probability density of the waiting time between successive arrivals.

5.63 Verify forα = 3 and β = 3 that the integral of the beta density, from 0 to 1, is equal to 1.

5.64 If the ratio of defective switches produced during com-plete production cycles in the previous month can be looked upon as a random variable having a beta distri-bution withα = 3 and β = 6, what is the probability that in any given year, there will be fewer than 5%

defective switches produced?

5.65 Suppose the proportion of error in code developed by a programmer, which varies from software to software, may be looked upon as a random variable having the beta distribution withα = 2 and β = 7.

(a) Find the mean of this beta distribution, namely, the average proportion of errors in a code from this engineer.

(b) Find the probability that a software developed by this engineer will contain 30% or more errors.

5.66 Show that whenα > 1 and β > 1, the beta density has a relative maximum at

x= α − 1 α + β − 2

5.67 With reference to the Example 19, find the probability that such a battery will not last 100 hours.

5.68 Suppose that the time to failure (in minutes) of certain electronic components subjected to continuous vibra-tions may be looked upon as a random variable having the Weibull distribution withα = 1

5andβ = 1 3. (a) How long can such a component be expected to

last?

(b) What is the probability that such a component will fail in less than 5 hours?

5.69 Suppose that the processing speed (in milliseconds) of a supercomputer is a random variable having the Weibull distribution withα = 0.005 and β = 0.125.

What is the probability that such a supercomputer will have similar processing speeds after running for 50,000 ms?

5.70 Verify the formula for the variance of the Weibull dis-tribution given on page 160.

5.10 Joint Distributions—Discrete and Continuous