Marginal and Conditional Distributions

Probability · Axiom Academy

LESSON Marginal and Conditional Distributions Extracting individual distributions and understanding conditional probabilities 1. Marginal Distributions from Joint PMF The marginal PMF of X is obtained by summing the joint PMF over all possible values of Y. In a joint PMF table, marginal probabilities are the row sums (for X) and column sums (for Y). They're called "marginal" because they appear in the margins of the table. The conditional PMF of X given Y = y describes the distribution of X when we know Y takes a specific value y. This is just Bayes' rule applied to random variables: we divide the joint probability by the marginal probability of the condition. Note: p Y (y) must be positive for the conditional PMF to be defined. To find marginals from a joint table: • Sum across each row to get p X (x) values • Sum down each column to get p Y (y) values • Fix a value of Y (select a column) • Divide each entry in that column by the column sum • Result: P(X = x | Y = y) for each x

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