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I have two independent variables $X$ and $Y$. $W=X-Y$ when $X\sim $Bernoulli$(1/2)$ and $Y\sim N(0,1)$. This puts $Var(x)=1/4$ and $Var(Y)=1$, but I have to be misunderstanding something because if $Var(W)=Var(X)-Var(Y)$ then the $Var(W)$ is negative, which makes no sense. What am I missing? |
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The thing you are misunderstanding is that $Var(X-Y)\not=Var(X)-Var(Y)$ |
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If $U$ and $V$ are independent, then $\text{Var}(aU+bV)=a^2\text{Var}(U)+b^2\text{Var}(V)$. In our case, the variance of $X-Y$, that is, of $(1)X+(-1)Y$, is the variance of $X$ plus the variance of $Y$. |
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Variance is not a linear mapping. When $X$ and $Y$ are independent (or more generally uncorrelated), $\operatorname{Var}(X+Y)=\operatorname{Var}(X)+\operatorname{Var}(Y)$. Here you should apply this noticing that $X-Y=X+(-Y)$ and $\operatorname{Var}(-Y)=\operatorname{Var}(Y)$. |
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