Independence and Expectation

When events don't influence each other, and the long-run average outcome.

Two events $A$ and $B$ are independent if knowing one happened tells you nothing about the other. Formally:

P(A \cap B) = P(A)\, P(B)

Equivalently, $P(A \mid B) = P(A)$ . Independence is a strong assumption — verify it, don't assume it.

The expectation of a random variable $X$ is its probability-weighted average. For discrete $X$ ,

E[X] = \sum_x x\, P(X = x)

For continuous $X$ with density $f$ ,

E[X] = \int_{-\infty}^{\infty} x\, f(x)\, dx

The single most useful property of expectation is linearity:

E[aX + bY] = a\, E[X] + b\, E[Y]

This holds whether or not $X$ and $Y$ are independent. Many problems that look impossible become easy once you write the answer as a sum of indicator variables and apply linearity.