Introduction to Probability and Statistics for Engineers and Scientists

Problems 609

and make the change of variables

x=αtβ, dx=αβtβ−^1 dt

24.Show that ifXis a Weibull random variable with parameters (α,β), then

Var(X)=α−2/β

[

(

1 +

2

β

)

−

(

(

1 +

1

β

)) 2 ]

25.If the following are the sample data from a Weibull population having unknown

parametersαandβ, determine the least square estimates of these quantities, using

either of the methods presented.

Data:15.4, 16.8, 6.2, 10.6, 21.4, 18.2, 1.6, 12.5, 19.4, 17

26.Show that ifXis a Weibull random variable with parameters (α,β), thenαXβis

an exponential random variable with mean 1.

27.IfU is uniformly distributed on (0, 1) — that is,Uis a random number —

show that [−(1/α) logU]1/β is a Weibull random variable with parameters

(α,β).

The next three problems are concerned with verifying Equations 14.5.5 and

14.5.7.

28.IfXis a continuous random variable having distribution functionF, show that

(a) F(X) is uniformly distributed on (0, 1);

(b) 1 −F(X) is uniformly distributed on (0, 1).

29.LetX(i)denoteith smallest of a sample of sizenfrom a continuous distribution

functionF. Also, letU(i)denote theith smallest from a sample of sizenfrom a

uniform (0, 1) distribution.

(a) Argue that the density function ofU(i)is given by

fU(i)(t)=

n!

(n−i)!(i−1)!

ti−^1 (1−t)n−i,0<t< 1

[Hint:In order for theith smallest ofnuniform (0, 1) random variables to

equalt, how many must be less thantand how many must be greater? Also,

in how many ways can a set ofnelements be broken into three subsets of

respective sizesi−1, 1, andn−i?]