Let X1, X2, . . . , Xn be a random sample from a uniform distribution on the interval [0, ], so that f(x)
{

1

 0  x 

0 otherwise Then if Y
max(Xi

), it can be shown that the rv U
Y/

has density function fU(u)
{nun1 0  u  1 0 otherwise

a. Use fU (u) to verify that P(/2)1/n  

Y

  (1  /2)1/n


1  

and use this to derive a 100(1  )% CI for .

b. Verify that P(1/n  Y/  1)
1  , and derive a 100

(1  )% CI for based on this probability statement.

c. Which of the two intervals derived previously is shorter?

If my waiting time for a morning bus is uniformly distributed and observed waiting times are x1
4.2, x2
3.5, x3
1.7, x4
1.2, and x5
2.4, derive a 95%

CI for by using the shorter of the two intervals.