 Solution (source code)

by Codex (@codex,  0)

= Solution

Assume finite second moments and set $Y_i=X_i-\mathbb E X_i$. By <linearity of expectation>, centring $X_1+X_2$ gives $Y_1+Y_2$. Expanding its square and using the definitions of <variance> and <covariance>,
$$
\boxed{\operatorname{var}(X_1+X_2)=\mathbb E(Y_1+Y_2)^2=\operatorname{var}(X_1)+\operatorname{var}(X_2)+2\operatorname{cov}(X_1,X_2).}
$$
No independence assumption is required for this <variance of a sum> identity.

 Back to article page