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2. Consider the following joint distribution of two discrete variables X and Y: fx,y(x, y) 01 2 3 お88 Recall that the marginal distribution of X is defined as: fx(x) and the marginal distribution of Y is defined as fy(v) -xf(i) Find fx(x) and fy(y) in the support of X and Y (or in simpler terms, find 1), P(Y = 0), P(Y-1), P(Y-2) and P(Y P(X-0), P(X 3))b. The conditional density of Y given X is given by: P(X=x) Find the conditional density of Y given X in the support of (X, Y) (or in simpler terms, find P(Y = 0 | X = 1), P(Y = 1 | X = 1), P(Y and P(Y 3 X-1). Same for conditioning on X 0). 2 | X = 1), , c. Recall that a sufficient condition for any two variables to be independent is: fx,Y(z,y) = fx (z)fy (y). Are X and Y independent, i.e. is P(X-, Y-P(X)P(Y y) Hint: Check this for values X-1, Y 3 d. Caciulate E(Y | X), ie. calculate E(Y | X = 0) and E(Y | X = 1). for all x, yE R? Is E(Y X) a constant or a random variable? e. Use the result from part d. to find the values of the parameters α and β for which E(y | X) is a linear function of X, ie, for values of α and β is E(Y | X) = α+BX? Hint: Find the values of the parameters α and β, such that E(Y | X)ecall that the Law of iterated expectations states: Find E(Y) using the Law of iterated expectations, 1.e . find E(Y) using the PMF of variable X. Hint: E(Y) = Ex[E(Y X)] = P(X = 0). E(Y X = 0) + P(X = 1). E(Y X = 1).

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