Let X and Y be discrete random variables with Sx = {0, 1, 2} and Sy = {0, 1, 2, 3}. Define the joint probability mass function f(x, y) = P(X = x, Y = y) as follows:
f(x, y) =
{
1/9 if x = 0, 1, 2 and y = 0, 1, 2, 3
0 otherwise
}
a) Find the marginal distribution of X. That is, find fX(x) = P(X = x) for x = 0, 1, 2. (Hint: fX(x) is the sum of the probabilities across the y values of f(x, y) for each value of x.)
b) Find the marginal distribution of Y. That is, find fY(y) = P(Y = y) for y = 0, 1, 2, 3. (Hint: fY(y) is the sum of the probabilities across the x values of f(x, y) for each value of y.)
c) Are X and Y independent random variables? (Hint: Check if f(x, y) = fX(x) * fY(y).)