c. Let X1, ..., X, be i.i.d. random variables from normal distribution with mean , and variance o? and Y1, . ... Ym be i.i.d. random variables from normal distribution with mean #2 and variance o2. Find the likelihood ratio test for testing Ho : /1 = #2, ? = 0? against all alternatives. The two samples are independent. d. Let X, , ..., Xjo be i.i.d. Bernoulli(p) random variables. Find the most powerful test of size a = 0.0547 for testing Ho : p = _ against H. : p = 1. Find the power of this test . e. Suppose that X , ..., X, are i.id. exponential random variables with EX, =$ t; where t, ..., t, are known constants, and > 0 is an unknown parameter . It can be shown the MLE off is given byB = 1 " Xi. (See also exam 2, problem 2c.) Suppose we want to test H. B = 1 against H . = 1. Find a test statistic given by the likelihood ratio test