7. Let the joint pmf of X and Y be defined by x+y x=1, 2.y=1,2,3,4 f (x,y)= 32 a) Find f(x), the marginal pmf of X. b) Find f(x), the marginal pmf of X.

. Calculate D(f ◦ g) for the functions f(x, y, z) = (x + y + z, x^3 − e^yz) and g(s, t, u) = (st, tu, su).

Suppose that a function f(x,y) has a critical point at (2,3). Given that fæx (2, 3) = -4, fæy(2,3) = 0, fyy(2, 3) = –5 what does the second derivative test tell us about the nature of this critical point?

Let X and Y have the joint pdf -1 f (x, y) =- c(x - y), 0≤ y ≤ x ≤ 1, 0, elsewhere a) Find c. b) Find P(X ≤ 2Y). c) Find the marginal pdf for X and Y.