Mean of a Random Variable

A fair die shows 1 through 6, each with probability 1/6. Multiply each value by its probability and add: 1(1/6)+2(1/6)+...+6(1/6) = 3.5. No roll ever lands on 3.5, but that's fine, because expected value isn't a prediction for one trial. It's the long-run average if you rolled the die thousands of times and kept a running tally.

Picture the probability histogram: each outcome is a bar, its height the probability. The expected value E(X) is the balance point of that histogram, the spot where it would balance on a fingertip if the bars were weights sitting on a seesaw. Values with more probability mass pull the balance point toward themselves, just like heavier weights pull a seesaw down on their side. That's why E(X) = sum of x times P(x): it's a weighted average, weighted by how likely each outcome is.

This idea is the foundation for everything that follows in the unit. The binomial distribution's mean, np, is just this formula applied to a specific counting situation. Later, when random variables get added or combined, rules for means build directly on this one: the mean of a sum is the sum of the means. Get comfortable with E(X) as a weighted average now, and those rules will feel like natural extensions instead of new things to memorize.