How to do Q3

consider the data to be a realization of Y, ., Y20 *** $(y; ), where Y, corresponds to the survival time for the ith patient and f is the exponential density: f(x;) = exp{-}, y>0 1. (1pt) Do you think the exponential density provides a reasonable model for the data relative to the other density functions shown below? Why or why not? Answer yes or no, and state one reason; limit your answer to two sentences. Exponential Gamma Gaussian Uniform 2. Consider = Y as an estimator of e. in (a) (1pt) Is biased for e? Show why or why not. (b) (1pt) is consistent for ? Show why or why not. (c) (1pt) is sufficient for o? Show why or why not. 3. The parts of this problem will guide you through constructing an exact confidence interval for 0 based on the point estimate Y using the pivotal method. (a) (2pt) Let X; = Y. Find the distribution of Xi. (b) (2pt) Find the distribution of Y = - X, using the MGF method and your ans the previous part. (C) (1pt) Show that Q= is a pivot for e. (Hint: see HW5, problem 1.) (d) (2pt) Find numbers a, b so that P(a a) = P(Q0 1. (1pt) Do you think the exponential density provides a reasonable model for the data relative to the other density functions shown below? Why or why not? Answer yes or no, and state one reason; limit your answer to two sentences. Exponential Gamma Gaussian Uniform 2. Consider = Y as an estimator of e. in (a) (1pt) Is biased for e? Show why or why not. (b) (1pt) is consistent for ? Show why or why not. (c) (1pt) is sufficient for o? Show why or why not. 3. The parts of this problem will guide you through constructing an exact confidence interval for 0 based on the point estimate Y using the pivotal method. (a) (2pt) Let X; = Y. Find the distribution of Xi. (b) (2pt) Find the distribution of Y = - X, using the MGF method and your ans the previous part. (C) (1pt) Show that Q= is a pivot for e. (Hint: see HW5, problem 1.) (d) (2pt) Find numbers a, b so that P(a a) = P(Q