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Chapter 5: Q 5.36 (page 214)

Suppose that the life distribution of an item has the hazard rate function位(t)=t3,t>0. What is the probability that

(a)the item survives to age2?

(b)the item鈥檚 lifetime is between.4and1.4?

(c)a 1-year-old item will survive to age2?

Short Answer

Expert verified

Therefore the,

(a)P(X鈮�2)鈮�0.0183(b)P(0.4鈮�X鈮�1.4)鈮�0.6109(c)P(X鈮�2X鈮�1)鈮�0.0235

Step by step solution

01

Given information:

The life distribution of an item has the hazard rate function位(t)=t3,t>0.

Let Xdenote the life distribution. First, we compute the probability function of the life distribution.

F(t)=1-exp-鈭�0t位(u)duF(t)=1-exp-14t4

02

Part (a) step 2 Explanation:

It is to computeP(X鈮�2)

P(X鈮�2)=1-P(X<2)=1-F(2)=e-4鈮�0.0183

03

Part (b) step 3 Explanation:

The probability is given by

P(0.4鈮�X鈮�1.4)=P(X鈮�1.4)-P(X鈮�0.4)=F(1.4)-F(0.4)=(1-e-0.9604)-(1-e-0.0064)=e-0.0064-e-0.9604鈮�0.6109

04

Part (c) step 4 Explanation:

The probability that a 1-year-old item will survive to age 2is

P(X鈮�2X鈮�1)=P(X鈮�2)P(X鈮�1)=1-F(2)1-F(1)=e-4e-14=e-154鈮�0.0235

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