Final answer:
The pmf of two added random variables, represented as Z = X + Y, is found by using the convolution of the individual pmfs of X and Y. This is calculated by summing the products of probabilities that yield each possible sum, considering all combinations of X and Y that contribute to that sum.
Step-by-step explanation:
The pmf of two added random variables is the probability mass function (pmf) that results when those two variables are summed. If X and Y are two discrete random variables, the pmf of their sum Z = X + Y can sometimes be computed using convolution. To calculate this, assume X takes on values x1, x2, ..., xn with probabilities p1, p2, ..., pn, and Y takes on values y1, y2, ..., ym with probabilities q1, q2, ..., qm. The pmf of Z=P(X+Y) can be found by summing the products of the probabilities of X and Y that result in a given sum.
For example, to find P(Z=z), you would calculate P(Z=z) = Σ P(X=xi)P(Y=z-xi), where the sum is taken over all xi such that (z - xi) is a value that Y can take on.
The central limit theorem can also be referenced when considering the distribution of the sum of random variables if we are dealing with the sum of a large number of independent, identically distributed variables.