Let $d \geq 1$ and $K \geq 1$ be integers. Let $D \subset \mathbb{R}^d$ be a Lebesgue-measurable set satisfying $0 < \mathcal{L}^d(D) < \infty$, and let $D_1,\dots,D_K \subset D$ be pairwise disjoint Lebesgue-measurable subsets such that $D = \bigcup_{k=1}^K D_k$ and $\mathcal{L}^d(D_k) > 0$ for every $k \in \{1,\dots,K\}$. For each $k$, let $\mathcal{D}_k$ denote the restriction of the Lebesgue $\sigma$-algebra on $\mathbb{R}^d$ to $D_k$. Define

\begin{align*} p_k := \frac{\mathcal{L}^d(D_k)}{\mathcal{L}^d(D)} \end{align*}

for $k \in \{1,\dots,K\}$.

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. For each $k \in \{1,\dots,K\}$, let $n_k \geq 1$ be an integer, and for each $j \in \{1,\dots,n_k\}$ let $X_{k,j} : (\Omega,\mathcal{F}) \to (D_k,\mathcal{D}_k)$ be a random vector with distribution $\operatorname{Unif}(D_k)$, meaning its law is the normalized [Lebesgue measure](/page/Lebesgue%20Measure) $\mathcal{L}^d(\cdot \cap D_k)/\mathcal{L}^d(D_k)$ on $(D_k,\mathcal{D}_k)$. Assume that the full finite family $\{X_{k,j} : 1 \leq k \leq K,\ 1 \leq j \leq n_k\}$ is independent. Let $g : (D,\mathcal{D}) \to (\mathbb{R},\mathcal{B}(\mathbb{R}))$ be measurable, where $\mathcal{D}$ is the restriction of the Lebesgue $\sigma$-algebra to $D$, and assume $g \circ X_{k,j} \in L^2(\Omega,\mathcal{F},\mathbb{P})$ for every $k,j$.

For each $k \in \{1,\dots,K\}$, define the stratum sample mean

\begin{align*} \bar{g}_k := \frac{1}{n_k}\sum_{j=1}^{n_k} g(X_{k,j}), \end{align*}

the stratified estimator

\begin{align*} \hat{I}_{\mathrm{strat}} := \mathcal{L}^d(D)\sum_{k=1}^K p_k \bar{g}_k, \end{align*}

and the stratum variance

\begin{align*} \sigma_k^2 := \operatorname{Var}(g(X_{k,1})). \end{align*}

Then

\begin{align*} \operatorname{Var}(\hat{I}_{\mathrm{strat}}) = \mathcal{L}^d(D)^2 \sum_{k=1}^K \frac{p_k^2\sigma_k^2}{n_k}. \end{align*}