Let $X_1, X_2, \dots$ be independent random variables, and show that $Y_n = X_n - \mathbb{E}[X_n]$ are independent
2026-03-27 21:19:35.1774646375
Prove expectation of independent R.V.s. are independent
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For $m\ne n$, $P(Y_n\le y_n,Y_m\le y_m)=P(X_n\le y_n+E(X_n),X_m\le y_m+E(X_m))$
$=P(X_n\le y_n+E(X_n))P(X_m\le y_m+E(X_m))=P(Y_n\le y_n)P(Y_m\le y_m)$