Question

2. (Total 7 points) Consider a continuous pair of random variables (X, Y) with joint PDF f_{X, Y}(x, y)=\frac{2}{5}(2 x+3 y) \text { for } 0 \leq x \leq 1 \text { and } 0 \leq y \leq 1 0 otherwise. \text { int) What is the marginal PMF of } Y, f_{Y}(y) ? \text { What are } E(Y) \text { and } \operatorname{Var}(Y) ? \text { t) What are } E(X Y) \text { and } \operatorname{Var}(X Y) ? \text { What is } \operatorname{Cov}(X, Y) ? \text { t) What is the conditional PDF of } X \text { given } Y \text {, i.e., } f_{X \mid Y}(x \mid y) ? \text { What is the conditional CDF of } X \text { given } Y \text {, i.e., } F_{X \mid Y}(x \mid y) \text { ? } \text { Calculate } E\left(X \mid Y=\frac{1}{2}\right) \text { and } \operatorname{Var}\left(X \mid Y=\frac{1}{2}\right) \text {. }

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