Section 14.1

Variance of Counts

Variances and correlations of indicator sums: hypergeometric variance, multinomial counts, E[XY] for heads and tails.

Toolkit

Write a count as a sum of indicators, $X=\sum_i \mathbf{1}_i$ , and expand variance/covariance via pairwise terms:

\operatorname{Var}(X)=\sum_i \operatorname{Var}(\mathbf{1}_i)+\sum_{i\ne j}\operatorname{Cov}(\mathbf{1}_i,\mathbf{1}_j),\qquad \operatorname{Cov}(\mathbf{1}_i,\mathbf{1}_j)=\mathbb{P}(i,j)-\mathbb{P}(i)\mathbb{P}(j).

Key cases:

Binomial: $\operatorname{Var}(X)=np(1-p)$ ; for $Y=n-X$ , $E[XY]=E[X]E[Y]+\operatorname{Var}(X)$ since $\operatorname{Cov}(X,Y)=-\operatorname{Var}(X)$ .
Multinomial counts $X_i\sim\mathrm{Bin}(n,p_i)$ with $\operatorname{Cov}(X_i,X_j)=-np_ip_j$ (same-trial indicators are mutually exclusive); use $\operatorname{Var}(X_i+X_j)$ as a check.
Hypergeometric (sampling without replacement from a finite population): indicator pairs give negative covariance, yielding
$\operatorname{Var}(X)=n p(1-p)\frac{N-n}{N-1}, \qquad p=\frac{K}{N}.$
Exchangeable 0/1 structures (e.g. permutation fixed points): use $\mathbb{P}(i)$ and $\mathbb{P}(i,j)$ directly to get pairwise covariances, then sum.