2. Simultaneous distribution: We consider two random variables X and Y with simultaneous probability distribution p (x, y) = P(X=xnY=y) given by this table: x/y 0 0 1 2 1 2 0.1 0.2 0.1 0 0.2 0.05 0.1 0.15 0.1 (a) Find the marginal probability distributions for X and Y. What is P (X= 2 | Y = 2)? Are X and Y independent random variables? Explain your answer. (b) Find the correlation between X and Y when you are told that the expectations are E (X) = 0.95 and E (Y) = 0.95, and the variances V (X) = 0.747 and V (Y) = 0.747. (c) Find the conditional expectation E (Y | X = 2) and the conditional variance V (Y | X = 2).

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