Given a joint PDF, f subscript X Y end subscript (x comma y )equals c x y comma space 0 less than y less than x less than 4, (1) (5 pts) Determine the constant c value such that the above joint PDF is valid. (2) (6 pts) Find P (X greater than 2 comma space Y less than 1 )(3) (9 pts) Determine the marginal PDF of X given Y

(1) Looks like the joint density is

In order for this to be a proper density function, integrating it over its support should evaluate to 1. The support is a triangle with vertices at (0, 0), (4, 0), and (4, 4) (see attached shaded region), so the integral is

(2) The region in which X > 2 and Y < 1 corresponds to a 2x1 rectangle (see second attached shaded region), so the desired probability is

(3) Are you supposed to find the marginal density of X, or the conditional density of X given Y?

In the first case, you simply integrate the joint density with respect to y:

In the second case, we instead first find the marginal density of Y:

Then use the marginal density to compute the conditional density of X given Y:

(x + 7) + (2x - 1) = 180, because the angles are on a straight line.

3x + 6 = 180

x + 2 = 60

x = 58

Hence, Angle FAE = (58 + 7)° = 65°.