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3. Probability and Statistics Review (30 points) - 3a. If (X, Y) follows a bivariate normal distribution with mean (,ul, ,ug) and covariance matrix 1,9 a1 ' Can you write down the joint distribution of (X, Y)? What is the conditional distribution of X given Y? What is the conditional expectation of X given Y? . 3b. Consider the following hypothesis testing problem H0:,u.=O vs leua. Our observation is a random variable X, which follow a normal distribution with mean ,u and variance 1. Naturally, we can choose T = X as our test statistics. If we would like to control Type I error at 0.05 level, how should we choose the critical value for T? Based on your choice of critical value, can you calculate the type II error if p = 5? Hint: you may want to use some function in dnorm, pnorm and qnorm. 0 3c. Suppose X1, . . . ,Xn follows Poisson distribution with parameter A. What is the limiting distribution of T? T is dened as X + + X = # _ ya. T a