Linearity of Expectation

Linearity of expectation states: $$E[X_1 + X_2 + \cdots + X_n] = E[X_1] + E[X_2] + \cdots + E[X_n]$$ This holds regardless of dependence. The $X_i$ do not need to be independent. Intuition: Expected value is a sum over outcomes weighted by probability. Sums distribute over sums. Power move: Break a complex random variable into simple indicator variables. The expected number of fixed points in a random permutation? Let $X_i = 1$ if element $i$ is fixed. Then $E[X] = \sum E[X_i] = n \cdot \frac{1}{n} = 1$. No need to compute the full distribution. When to use: Almost every expected value problem. If you're computing a distribution when you only need the mean, you're probably overcomplicating it. This is a fundamental technique. Use it freely. It's the single most useful technique in discrete probability.