Introduction to Probability and Statistics for Engineers and Scientists

Problems 605

4. Suppose the life distribution of an item has failure rate functionλ(t)=t^3 ,0<
   t<∞.
   (a) What is the probability that the item survives to age 2?
   (b) What is the probability that the item’s life is between .4 and 1.4?
   (c) What is the mean life of the item?
   (d) What is the probability a 1-year-old item will survive to age 2?


5. A continuous life distribution is said to be an IFR (increasing failure rate)
   distribution if its failure rate functionλ(t) is nondecreasing int.
   (a) Show that the gamma distribution with density

f(t)=λ^2 te−λt, t> 0

is IFR.

(b) Show, more generally, that the gamma distribution with parametersα,λis

IFR wheneverα≥1.

Hint:Write

λ(t)=

[∫∞

t λe

−λs(λs)α− (^1) ds
λe−λt(λt)α−^1
]− 1

6. Show that the uniform distribution on (a,b) is an IFR distribution.


7. For the model of Section 14.3.1, explain how the following figure can be used to
   show that

τ=

∑r

j= 1

Yj

where

Yj=(n−j+1)(X(j)−X(j−1))

0

X(r)

123 r − 3 r − 2 r − 1 r n

X(r − 1)

X(r − 2)

X(3)

X(2)

X(1)

(Hint:Argue that bothτand

∑r

j= 1 Yjequal the total area of the figure shown.)