Discrete Probability Distributions

• E(X) — the expected value (mean) of a discrete random variable: multiply each value by its probability and add them up.
• E(X²) and E(g(X)) — the expected value of a function of X, found by applying the function to each x-value first, then weighting by probability.
• Var(X) — the variance (spread) of X, calculated as E(X²) − (E(X))².
• Linear transformations — how E(X) and Var(X) change when you do aX + b to every value of X.
• Problem solving — building your own distribution tables from word problems, combining two independent variables, and handling a variable Y that depends on X.

E(X) stands for the expected value — but that name is a little misleading, because it doesn't mean "the value you should expect to see." It's really the long-run average if you repeated the random experiment thousands of times and averaged your results.

Here's the classic example that makes this click: if you flip a fair coin 5 times, the expected number of tails is 2.5. You will never actually get 2.5 tails on a single go — you'll get 0, 1, 2, 3, 4, or 5. But if you did this experiment thousands of times and averaged all your results, you'd land right around 2.5. E(X) doesn't have to be an achievable outcome — it's a theoretical balancing point.

Think of it like a physical seesaw. Imagine each value of X sitting on a number line, and its probability is how "heavy" that value is. E(X) is exactly where you'd need to place the pivot so the whole seesaw balances perfectly. Values with bigger probabilities pull the balance point more strongly toward them.

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