2. Let X and Y be continuous random variables with joint p.d.f. Find the f(x, y) = k(x + y), 0≤x≤y≤1 (a) value of the constant k (b) joint c.d.f of X and Y F(x, y) marginal c.d.f. of X and marginal c.d.f. of Y. marginal p.d.f. of X and marginal p.d.f. of Y.

Let X and Y be continuous random variables with joint p.d.f. f(x, y) = k(x + y), 0≤x≤y≤1 Find the (a) (b) (c) (d) (a) value of the constant k joint c.d.f of X and Y F(x, y) marginal c.d.f. of X and marginal c.d.f. of Y. marginal p.d.f. of X and marginal p.d.f. of Y. Given that X is a continuous r.v. with p.d.f. kx-(0+1) x>α; 0>0, a>

Consider the pair of random variables ( X , Y ) with joint PDF f X , Y ( x , y ) = 2 for 0 ≤ x ≤ y ≤ 1 and f X , Y ( x , y ) = 0 otherwise. (In particular, note that the joint PDF is only nonzero when x ≤ y .) (a) Draw a picture of the range of ( X , Y ) in the x y -plane. (b) Describe the conditional distribution of X given Y = y , for 0 ≤ y ≤ 1 . (c) What is the marginal PDF f X of X ? (d) Are X and Y independent?

2. Suppose that X and Y are independent random variables, both with the standard normal distribution i.e. X, Y~ N(0, 1). For pe [-1, 1], define Z:=√1-p²X + pY. Show that E[Z|Y] =pY.