Final answer:
To find the value of c for the function to be a joint PMF, we ensure all probabilities are non-negative and their sum equals 1, leading to the conclusion that c must be 1/14.
Step-by-step explanation:
To determine the value of c such that the function f(x, y) = cy(2y - x) serves as a joint probability mass function (PMF) for two discrete random variables, we must ensure that the PMF satisfies two conditions:
-
- All probabilities must be non-negative.
-
- The sum of all probabilities must be equal to 1.
Given that x can take the values 0 or 3, and y can take the values 0, 1, or 2, we first check if f(x, y) is non-negative for all these value combinations. It is, as y and (2y - x) are either zero or positive for the given values of x and y. Next, we calculate the sum of probabilities for all possible pairs of (x, y):
Σf(x, y) = f(0,0) + f(0,1) + f(0,2) + f(3,0) + f(3,1) + f(3,2)
Inserting the actual function into the equation:
c(0)(2*0 - 0) + c(1)(2*1 - 0) + c(2)(2*2 - 0) + c(0)(2*0 - 3) + c(1)(2*1 - 3) + c(2)(2*2 - 3)
Which simplifies to:
c * (0 + 2 + 8 + 0 + (-1) + 5)
Therefore, we have:
c * 14 = 1
To find c, we solve the equation:
c = 1/14
Thus, c must be 1/14 for f(x, y) to serve as a legitimate PMF.