Expectation and Variance of the Binomial and Geometric Distributions

Core Concept

The Binomial Distribution

If $X \sim \text{B}(n, p)$, then $X$ counts the number of successes in $n$ independent trials, each with probability of success $p$ (and failure $q = 1-p$).

The expectation $\text{E}(X) = np$ reflects the intuitive idea that, across $n$ trials, you expect a fraction $p$ to succeed. The variance $\text{Var}(X) = npq$ captures how much the count fluctuates — it is largest when $p = 0.5$ (maximum uncertainty) and shrinks as $p$ approaches 0 or 1.

The Geometric Distribution

If $X \sim \text{Geo}(p)$, then $X$ counts the number of trials needed to obtain the first success, so $X \in \{1, 2, 3, \ldots\}$.

The expectation $\text{E}(X) = \dfrac{1}{p}$ is deeply intuitive: if each trial has a $\tfrac{1}{5}$ chance of success, you expect to wait about 5 trials for the first success.

Syllabus note: The 9709 syllabus requires the expectation formula for the geometric distribution only. The variance formula $\text{Var}(X) = \dfrac{q}{p^2}$ exists but is outside the scope of this objective — do not spend time on it unless a question explicitly provides it.