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In probability theory, when two discrete random variables are considered together, their combined behaviour is described using the joint probability mass function. It provides information about the probability that two random variables take specific values simultaneously. Let X and Y be two discrete random variables defined on the same sample space. The joint probability mass function (joint PMF) of X and Y is denoted by p(x, y) and is defined as p(x, y) = P(X = x, Y = y) where x and y represent the possible values of X and Y. The joint PMF gives the probability that X assumes the value x and at the same time Y assumes the value y. Since probabilities cannot be negative, p(x, y) ≥ 0 for all values of x and y. Also, the sum of all joint probabilities over the entire range of X and Y must be equal ________ ____ ____ ___ ______ _____ _________ _______ ______ _________.
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