Determine the value of c that makes fxyx y cx y a valid

Question

Question 1. Determine the value of c that makes f_XY(x, y) = c(x + y) a valid joint PDF over the range of 0 < x < 2, 0 < y < 2.

Question 2. Given a joint PDF f_XY(x, y) = 10e^-2x-3y over the range 0 < y < x < ∞,

1) [True or False] f_x(x) = 10∫₀^xe^{-2x-3y dy}

Write the correct integral to calculate f_x(x) if you think it is False:

2) [True or False] P(X < 2, Y < 1) = 10 ∫₀¹∫₀²e^-2x-3ydx dy

Write the correct integral to calculate P(X < 2, Y < 1) if you think it is False:

3) Verify that the given joint PDF is a valid.

Question 3. Given joint PMF of X and Y are given as follows. Determine the marginal PMF of Y.

X	Y	FXY(x, y)
0	0	0.5
1	0	0.25
2	1	0.25