4. Let X and Y have the joint probability distribution function
f(x,y) = { c(x^2 + y) for x ∈ {1, 2}, y ∈ {1, 2}; 0 otherwise }
a. Find the constant c.
b. Find the correlation between X and Y.
c. Calculate E[Y|X = x]
5. Let X and Y be independent random variables and have the following probability density function:
f_x(x) = f_y(y) = { e^-x for x > 0; 0 otherwise }
Find probability density functions of U = max(X, Y) and V = min(X, Y).
6. Let X be the random variable with the following density function
f(x) = { 2x for 0 < x < 1; 0 otherwise }
Find the probability density functions of the random variables Y = 3X + 1, Z = -3X + 1.
7. Let X and Y be two random variables with the following joint probability density function:
f(x,y) = { cx^2y for x^2 < y < 1; 0 otherwise }
a. Find the constant c.
b. Find the correlation between X and Y.
c. Find P(X ≥ Y)
d. Find the marginal pdfs of X and Y, respectively.
e. Find P(Y ≥ 3/4 | X = 0.5).
e. Find E[Y|X = x].