Final answer:
The constant C is found to be 1/28 to satisfy the total probability theorem. The probability P[Y>X] is 3/28, and X and Y's marginal PMFs can be computed by summing up the joint probabilities. X and Y are not independent since the joint PMF is not the product of the marginals.
Step-by-step explanation:
Finding the Constant C and Probabilities for Joint PMF
To find the constant C we must ensure that the sum of all probabilities equals 1. We have the joint PMF Px.y(x,y) = cxy at points x = 1,2,4 and y = 1,3 and zero otherwise.
The sum of Px.y(x,y) over all values of x and y will equal 1:
1 * C * 1 + 1 * C * 3 + 2 * C * 1 + 2 * C * 3 + 4 * C * 1 + 4 * C * 3 = 1
C(1 + 3 + 2 + 6 + 4 + 12) = 1
C(28) = 1
C = 1/28
To find P[Y>X], we look at the pairs (x, y) where y is greater than x:
(1, 3). So P[Y>X] = (1/28) * (1 * 3) = 3/28.
The marginal PMFs of X and Y are found by summing the joint PMF over the appropriate variable Y and X respectively. For X:
P(X=1) = Px.y(1,1) + Px.y(1,3) = C(1*1) + C(1*3) = 4C
P(X=2) = Px.y(2,1) + Px.y(2,3) = C(2*1) + C(2*3) = 8C
P(X=4) = Px.y(4,1) + Px.y(4,3) = C(4*1) + C(4*3) = 16C
And similar calculations are performed for Y.
Finally, we determine if X and Y are independent. Random variables X and Y are independent if for all x and y, P(X=x AND Y=y) = P(X=x)P(Y=y). In this case, since the joint PMF is a product of x and y times a constant and does not factor into a product of a function of x and a function of y, X and Y are not independent.