Final answer:
The value of c that makes the joint pmf valid is obtained by summing all possible values of fX,Y(x,y) and setting it equal to 1. P[Y<=X] can be evaluated by summing the probabilities of pairs (x, y) where y is less than or equal to x. The expressions for fX(x) and fY(y) can be found by considering the probabilities of pairs (x, y) where x or y takes on a specific value.
Step-by-step explanation:
The given joint pmf is: fX,Y(x,y) = c/x2 for x = 1,2 and y = 1,2,3,4,5,6. In order for this to be a valid joint pmf, the sum of all possible values of fX,Y(x,y) must equal 1. So, we need to find the value of c that satisfies this condition.
P[Y<=X] can be evaluated by finding the sum of the probabilities of all the pairs (x, y) such that y is less than or equal to x. In this case, we have P[Y<=X] = fX,Y(1,1) + fX,Y(2,1) + fX,Y(2,2).
To find the expression for fX(x), we need to sum up all the probabilities of the pairs (x, y) where x takes on a specific value. In this case, we have fX(1) = fX,Y(1,1), fX(2) = fX,Y(2,1) + fX,Y(2,2).
The expression for fY(y) can be found by summing up all the probabilities of the pairs (x, y) where y takes on a specific value. In this case, we have fY(1) = fX,Y(1,1), fY(2) = fX,Y(1,2) + fX,Y(2,1), fY(3) = fX,Y(2,3), fY(4) = fX,Y(2,4), fY(5) = fX,Y(2,5), fY(6) = fX,Y(2,6).
To evaluate Cov[X, Y], we need to first find E[X], E[Y], E[XY], and then use the formula Cov[X, Y] = E[XY] - E[X]E[Y].