🗒️ Ben's Notes

Expectation and Variance

The expectation of a random variable, $E(X)$, is the average of possible values weighted by their probabilities. Formally, it can be defined in two ways:

  1. Domain definition: $E(X) = \sum_{\omega \in \Omega} X(\omega) P(\omega)$.
  2. Range definition: $E(X) = \sum_x x P(X = x)$.

Expectation has nice properties of linearity: $E(X + Y) = E(X) + E(Y)$ and $E(aX + b) = aE(x) + b$.