Expectations and Variances of Independent Random Variables

In the setting of Probability Spaces: Notation, let $n\in\mathbb{N}$.

(i) Expectation of a product. Let $X$ and $Y$ be independent random variables. If $X\ge0$ and $Y\ge0$, then, with $X$, $Y$ and $XY$ as elements of $\mathcal{M}^{+}(\Omega,\mathcal{F})$ (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (vi)),

$$\mathbb{E}[XY]=\mathbb{E}X\cdot\mathbb{E}Y\qquad\text{in }[0,\infty].$$

If $X$ and $Y$ are integrable, then $XY$ is integrable and

$$\mathbb{E}[XY]=\mathbb{E}X\,\mathbb{E}Y.$$

(ii) Independent random variables are uncorrelated. Let $X,Y\in\mathcal{L}^{2}(P)$ be independent. Then $X$ and $Y$ are uncorrelated:

$$\operatorname{Cov}(X,Y)=0.$$

(iii) Variance of a sum. Let $X_{1},\dots,X_{n}\in\mathcal{L}^{2}(P)$ be such that $X_{j}$ and $X_{k}$ are uncorrelated for all $j,k\in[n]$ with $j\neq k$. Then $\sum_{k=1}^{n}X_{k}\in\mathcal{L}^{2}(P)$ and

$$\operatorname{Var}\Bigl(\sum_{k=1}^{n}X_{k}\Bigr)=\sum_{k=1}^{n}\operatorname{Var}X_{k}.$$

(iv) Covariance matrix of a sum. Let $X,Y\in\mathcal{L}^{2}(P;\mathbb{R}^{n})$ be independent random vectors. Then $X+Y\in\mathcal{L}^{2}(P;\mathbb{R}^{n})$ and

$$\operatorname{Cov}(X+Y)=\operatorname{Cov}(X)+\operatorname{Cov}(Y).$$