Final answer:
The joint pmf of x and y, given by f(x, y) = xy/32, can be used to calculate the probability of any pair of x and y by substitution. To confirm if this function represents a valid joint pmf, one must check that the sum over all possible pairs of x and y equals 1, which ensures a valid probability distribution.
Step-by-step explanation:
The question asks us to find the joint pmf (probability mass function) of two discrete random variables x and y, where their joint pmf is given by f(x,y) = xy/32. To find the joint pmf, one must ensure that the sum over all possible values of x and y equals 1, which constitutes a valid probability distribution.
For example, if x and y take integer values from 1 to 4, we would verify that ΣΣ(xy/32) over all x and y in the given range equals 1. If the function represents a valid joint pmf, then values such as P(x=2, y=3) are found directly by substituting x and y into the joint pmf function: P(x=2, y=3) = (2*3)/32 = 6/32. Furthermore, individual pmfs for x and y can be found by summing over the other variable, for instance, P(x) = Σ(y=1 to 4) xy/32. This should be done for all values x might take.