Discrete Distributions

Algebra 2 · Axiom Academy

A probability mass function gives every value of a discrete random variable a probability — and its expected value is the long-run average. 1. Random Variables and the PMF A discrete random variable X takes specific numeric values, and each value x has a probability P(X = x) . The full table (or rule) listing those probabilities is the probability distribution , or probability mass function (PMF). For two fair coin flips the four equally likely outcomes — — each map to a number of heads. Because HT and TH both give one head, the value X = 1 collects two outcomes and carries twice the probability. 2. What Makes a Distribution Valid A list of probabilities is a valid distribution only when it obeys two rules: every probability is between 0 and 1 , and together they account for all of the probability — they sum to exactly 1 . Each bar is between 0 and 1 , and stacking them leaves no gap and no overflow: The expected value E(X) , written , is the probability-weighted mean — the long-run average if you repeated the experiment many times. Multiply each value by its probability, then add: On average, two flips give one head — which here is also a value X can take. The expected value need not be attainable: a die never shows 3.5 , yet that is its long-run average roll. Once you have the PMF you answer probability questions by adding bars . For a single value, read one bar: . For a range, add every bar in the event — for example combines the bars at 1 and 2 .

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