2. A bivariate probability density function is called copula if its marginal probability density functions are uniform random variables over the interval (0, 1). For a constant α

between −1 and 1, let X and Y be continuous random variables with the joint probability density function given by f(x, y) = 1 + α(1 − 2x)(1 − 2y), 0 < x < 1, 0 < y < 1.

The function f is an example of a copula bivariate probability density function. Copula distributions have significant applications in portfolio optimization and financial risk assessment. Find fX|Y (x|y) and E(X | Y = y).