Let X1, X2, ….., Xn be a random sample from a uniform distribution on the interval [0,...

Let X₁, X₂, ….., X_n be a random sample from a uniform distribution

on the interval [0, θ], so that

Then if , it can be shown that the rv U = Y/θ has density function

a. Use f_U(u) to verify that

and use this to derive a 100(1 –α)% and CI for θ

b. Verify that , and derive a 100(1 –α)% CI for θ % 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 x = 2.4, derive a 95% CI for θ by using the shorter of the two intervals.