Joint Distributions

Probability · Axiom Academy

Describing the probability distribution of multiple random variables together 1. Joint PMF for Two Variables The joint probability mass function for discrete random variables X and Y gives the probability that X = x AND Y = y occur together. This extends naturally to more than two variables: p(x, y, z) = P(X = x, Y = y, Z = z). The joint PMF must satisfy: p(x, y) ≥ 0 and ΣₓΣᵧ p(x, y) = 1 (summing over all possible pairs). For discrete variables, we often represent the joint PMF as a table where rows correspond to values of X, columns to values of Y, and each cell contains p(x, y). Reading the table: each entry shows the probability of that specific combination of X and Y values. The sum of all entries in the table must equal 1, since the table covers all possible outcomes. 3. Independence of Random Variables Two random variables X and Y are independent if knowing the value of one gives no information about the other. In other words, the joint probability factors into the product of individual (marginal) probabilities. If this factorization doesn't hold for any pair (x, y), then X and Y are dependent.

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