Discrete Random Variables and Their Distributions

A discrete random variable is a number attached to the outcome of a random process: the sum you roll on two dice, the number of heads in five flips, the number of defective parts in a batch. Its probability distribution is just a table or graph pairing each possible value with its probability. Picture a histogram where the bar heights are probabilities instead of frequencies. Since the random variable has to take one of its listed values, every single time, those bar heights must add up to 1. That single fact, probabilities sum to 1, is what lets you find any missing probability in a distribution: subtract everything you know from 1.

Building a distribution means listing outcomes systematically, then counting. For the sum of two fair dice, there are 36 equally likely ordered outcomes, from (1,1) to (6,6). Group them by their sum: only (1,1) gives 2, so P(2) = 1/36, while six combinations give 7, so P(7) = 6/36. Lay out every sum from 2 to 12 this way and the probabilities land in a symmetric pyramid shape, peaking at 7, and they total exactly 1.

This topic is the bridge between the two-way tables and simulations earlier in the unit and the named distributions coming next, like the binomial. Every binomial or geometric distribution is still just a discrete random variable with a table of value-probability pairs; the formulas you'll meet later are shortcuts for building that table instead of a brand-new idea. Getting comfortable now with sum-to-1 reasoning, and with computing means and variances from a distribution, makes those later formulas feel like special cases rather than new rules.