Question

8. Suppose W and Z have a bivariate normal distribution 1 1 2(1-p2) (-2pzw+u2) fzw (z, w) 27T 1 (i) Find the marginal density of W then compute its MGF Mw and use it to find the mean and the variance of W. [3 (ii Find fzw (z|w) and use it to identify the distribution of Z given W = w aW bwhere a, b E R. [2 (iii) Derive the density of Y (iv) Compute the mean and variance of Y. You can use any general properties of the mean and variance you are familiar with. [2 (v) Find E(Z|W = w). Explain your derivation. Hint: you can use the previous results. 2 (vi) Find the covariance of Z and W. Explain your derivation. [2] (vii What is the best MSE predictor of Z given W? You are allowed to use what you learned about the general MSE predictor. 2]

Answer 1

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