Let f (x, y) = c, 0 ≤ y ≤ 4, y ≤ x ≤ y...

Let f (x, y) = c, 0 ≤ y ≤ 4, y ≤ x ≤ y + 1, be the joint pdf of X and Y.

(a) (3 pts) Find c and sketch the region for which f (x, y) > 0.

(b) (3 pts) Find f_X(x), the marginal pdf of X.

(d) (3 pts) Find P(X ≤ 3 − Y).

(e) (4 pts) E(X) and Var(X).

(f) (4 pts) E(Y) and Var(Y).

(g) (3 pts) Cov(X,Y).

(h) ( 3 pts) Find ρ, the correlation coefficient of X and Y.

(i) (3 pts) Are X and Y independent or dependent? Why or why not?

(j) (3 pts) Determine h(y | x), the conditional pdf of Y, given that X = x.

(k) (3 pts) Determine g(x | y), the conditional pdf of X, given that Y = y.

(l) (3 pts) Compute E(Y | x), the conditional mean of Y, given that X = x.

(m) (3 pts) Compute E(X | y), the conditional mean of X, given that Y = y.

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Expert Solution

orchestra answered 1 year ago

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