Let X1, X2,..., Xn be a random sample from a uniform distribution on the interval [0, ø], so that

Then if Y = max (Xi), it can be shown that the rv U = Y/ø has density function

a. Use fU(u) to verify that

and use this to derive a 100(1 - α)% CI for ø.
b. Verify that P(α1/n ‰¤ Y/ ø ‰¤ 1) = 1 - α, and derive a 100(1 - α)% CI for u 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 u by using the shorter of the two intervals.

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