Question

Let x1, . . . , xn be a random sample from a uniform distribution on the interval [0, θ], so that f (x; θ) = { 1/θ, 0 ≤ x ≤ θ 0, otherwise (1) then if y = x(n), it can be shown that the rv u = y /θ has density function fu (u; θ) = { nun−1, 0 ≤ u ≤ 1 0, otherwise (2) (a) use fu (u) to verify that p ( (α/2)1/n < y /θ ≤ (1 − α/2)1/n ) = 1 − α and use this to derive a 100(1 − α)% ci for θ. (b) verify that p ( α1/n < y /θ ≤ 1 ) = 1 − α and derive a 100(1 − α)% ci for θ 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, x5 = 2.4, derive a 95% ci for θ by using the shorter of the two intervals

Answer 1

To derive a confidence interval for theta from a uniform distribution sample, integrate the density function over the interval dictated by the given probability statement. The maximum observed sample value is used to evaluate the interval, with preference for the shorter interval for greater precision while keeping confidence levels.

Step-by-step explanation:

When we have a sample x1, ..., xn from a uniform distribution on the interval [0, θ] and y = x(n), the random variable (RV) u = y /θ has the density function fu (u; θ). To verify the probability statement P ( (α/2)^{1/n} < y /θ ≤ (1 - α/2)^{1/n} ) = 1 - α and derive a 100(1 - α)% confidence interval (CI) for θ, we can integrate the density function over the given interval. The same process applies for verifying P ( α1/n < y /θ ≤ 1 ) = 1 - α and constructing the corresponding CI.

In the given example, where the waiting time for a morning bus is observed and uniformly distributed with observed waiting times as x1 = 4.2, x2 = 3.5, x3 = 1.7, x4 = 1.2, x5 = 2.4, we would use the maximum observed waiting time to evaluate y, which is y = 4.2. For a 95% CI, we would use the shorter of the two previously derived intervals. We typically prefer the shorter interval because it provides a more precise estimate of the parameter θ, while still maintaining the desired level of confidence.