The random variables X,Y have joint probability density function

f(x, y) = c(x + y) for 0x1 0y1

= 0 for all other (x, y)

(a) Find the value of c for which this is a proper probability density function.

(b) Find the conditional probability density function of Y, given X = 0.5.

(c) Find the conditional expected value and conditional variance of Y , given X = 0.5.

(d) Compute the correlation coefficient between X and Y and show whether the random variables X,Y are independent.