I have three questions shown in the following screenshots

Problem 2. Let X1, X2, . . ,Xn be i.i.d. random samples (for example, the times of failure of 12. items) from an exponential distribution With PDF f(;r) = AsaA\" for a: 2 0. We want to construct a condence interval for i which is the expected value of the expo nential distribution. (a) (5 points) Show that Z : 2A 223:1 X,- has a chisquared With 2n degrees of freedom. (Hint: Compare moment generating functions.) (b) (3 points) Show that 2 2?:1 Xi 2 2:21 Xi Xingu/2 ' X2n,1o:/2 3 is a (1 a)level condence interval for i (c) (2 points) Suppose 10 identical batteries are tested for the number of hours of con tinuous operation With the following results: 4.6, 5.8, 10.6. 12.5, 13.4, 14.3, 20.6, 21.9, 22.1, 23.0. Assuming exponential distribution for the failure times, nd a 95% 1 2 HOMEWORK 3 condence interval for the mean lifetime of batteries. Problem 6. (a) (5 points) Let X be a random variable with a continuous cumulative distribution function F (so that the quantile function 17"1 is well-dened). Dene a new random variable Y : F (X ) Show that Y has a uniform distribution on the interval [0,1]. HOMEWORK 3 3 (b) (5 points) Let X1, X2, . . . ,Xn be i.i.d. random samples from a N(,u, 02) distribution, with known 02. Consider testing H0:p=0 versus H1:p>0 based on the test statistic Z : fig. For observed data 3:1, 302, . . . ,xn the pvalue is p : 1 @(z), with z : %. Thus, over repeated random samples with the same sample size n, the pvalue P:1cI>(Z) is a random variable. Use part (a) to show that P has a uniform distribution on the interval [0,1], and use this result to further show that a test which rejects H0 when the pvalue is less than a has the probability of making Type I error equal to 0:. Problem 7. Let X1,X2,...,Xn be i.i.d. random samples from N(,u,02). Consider the problem of testing 55:02:03 and H1:02>03, at signicance level :1, using the test that rejects H0 if (n US? 0(2) > Xiill Where 82 is the sample variance. a (3 points) Find an expression for the power of this test in terms of the xi, distri bution if the true 02 = (:03, Where c > 1. (b) (2 points) Find the power of this test if a : 0.05, n : 16, and c : 4, that is, the true 02 is four times the value being tested under H0