How do I show that: $$ E[(X − E[X] + Y − E[Y])]^2] \leq E[2(X − E[X])^2 + 2(Y − E[Y])^2] $$
2026-04-01 07:41:37.1775029297
Simple inequality with Expectations
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The inequality is equivalent to showing $$ \text{Var}(X+Y)\leq 2\text{Var}(X)+2\text{Var}(Y)\tag{1} $$ Note that $$ \text{Var}(X+Y)=\text{Var}(X)+\text{Var}(Y)+2\text{Cov}(X,Y) $$ whence (1) is equivalent to showing $$ \text{Var}(X)+\text{Var}(Y) -2\text{Cov(X,Y)}\geq 0.$$ But this is clear since $$ \text{Var}(X-Y)=\text{Var}(X)+\text{Var}(Y)-2\text{Cov}(X,Y) $$ and variance is always non-negative.