Probability Mass Functions

Probability · Axiom Academy

LESSON Probability Mass Functions Describing the probability distribution of discrete random variables 1. PMF Definition and Notation The Probability Mass Function (PMF) of a discrete random variable X gives the probability that X takes each specific value. For each possible value x, the PMF tells us p(x), the probability that the random variable equals x. The PMF completely characterizes the distribution. Every valid PMF must satisfy two essential properties: Property 1 (Non-negativity): p(x) ≥ 0 for all values x. Probabilities cannot be negative. Property 2 (Sums to 1): The sum of all probabilities equals 1. That is, Σ p(x) = 1 over all possible values of x. We typically visualize a PMF using a bar chart or stem plot, where the height of each bar represents the probability. The x-axis shows the possible values of the random variable, and the y-axis shows their probabilities. This makes it easy to see which values are more likely. Example: For a fair die, each value 1, 2, 3, 4, 5, 6 has probability 1/6, so all bars have equal height.

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