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Chapter 5: Q. 5.23 (page 216)

Compute the hazard rate function of a Weibull random variable and show it is increasing when 饾浗鈮�1and decreasing when 饾浗鈮�1

Short Answer

Expert verified

The hazard rate function of Weibul distribution is H(t)=饾浑t饾浗

Step by step solution

01

Given Information

X is a Weibull distribution with parameters 饾泴,饾泜and饾灚

02

Calculations

The hazard function of Weibull function is

H(t)=饾浑t饾浗

03

Plot

In order to better understand we need to see the graph of above function that is

where green graph is for the case when 饾浗鈮�1and black graph is for the case when localid="1650453593374" 饾浗>0.

This proves the statement thathazard rate function of a Weibull random variable and show it is increasing whenand decreasing when

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Most popular questions from this chapter

Suppose that X is a normal random variable with

mean 5. If P{X > 9} = .2, approximately what is Var(X)?

Use the result that for a nonnegative random variableY,

EY=鈭�0鈭�PY>tdt

to show that for a nonnegative random variableX,

EXn=鈭�0鈭�nxn-1PX>xdx

Hint: Start with

EXn=鈭�0鈭�PXn>tdt

and make the change of variablest=xn.

Let Xbe a random variable that takes on values between0andc. That isP0鈮�X鈮�c=1.Show that

Var(X)鈮�c24

Hint: One approach is to first argue that

localid="1646883602992" EX2鈮�cEX

and then use this inequality to show that

Var(X)鈮�c2[伪(1-伪)]where伪=E[X]c

The annual rainfall in Cleveland, Ohio, is approximately a normal random variable with mean 40.2 inches and standard deviation 8.4 inches. What is the probability that (a) next year鈥檚 rainfall will exceed 44 inches? (b) the yearly rainfalls in exactly 3 of the next 7 years will exceed 44 inches? Assume that if Ai is the event that the rainfall exceeds 44 inches in year i (from now), then the events Ai, i 脷 1, are independent.

Compute the hazard rate function of a gamma random variable with parameters (伪,位)and show it is increasing when 伪鈮�1and decreasing when 伪鈮�1
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