Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

4.10.∗Tolerance Limits for Distributions 319

EXERCISES

4.10.1.LetY 1 andYnbe, respectively, the first and thenth order statistic of a
random sample of sizenfrom a distribution of the continuous type having cdfF(x).
Find the smallest value ofnsuch thatP[F(Yn)−F(Y 1 )≥ 0 .5] is at least 0.95.

4.10.2.LetY 2 andYn− 1 denote the second and the (n−1)st order statistics of
a random sample of sizenfrom a distribution of the continuous type having a
distribution functionF(x). ComputeP[F(Yn− 1 )−F(Y 2 )≥p], where 0<p<1.

4.10.3.LetY 1 <Y 2 <···<Y 48 be the order statistics of a random sample of size
48 from a distribution of the continuous type. We want to use the observed interval
(y 4 ,y 45 ) as a 100γ% tolerance interval for 75% of the distribution.

(a)What is the value ofγ?

(b)Approximate the integral in part (a) by noting that it can be written as a par-

tial sum of a binomial pdf, which in turn can be approximated by probabilities

associated with a normal distribution (see Section 5.3).

4.10.4.LetY 1 <Y 2 <···<Ynbe the order statistics of a random sample of size
nfrom a distribution of the continuous type having distribution functionF(x).

(a)What is the distribution ofU=1−F(Yj)?

(b)Determine the distribution ofV =F(Yn)−F(Yj)+F(Yi)−F(Y 1 ), where

i<j.

4.10.5.LetY 1 <Y 2 <···<Y 10 be the order statistics of a random sample from
a continuous-type distribution with distribution functionF(x). What is the joint
distribution ofV 1 =F(Y 4 )−F(Y 2 )andV 2 =F(Y 10 )−F(Y 6 )?