I have the following solved exercise:
Let $X_1, X_2$ be independent random variables with $Xi \sim (1,1)$ and let $Z_1, Z_2$ be also independent random variables with $Zi \sim (0,1)$, find the distribution of $W = \frac{(\bar X - \bar Y)} { 2 }$
As this is a linear combination of two independent random variables then:
$W \sim N (E(W), V(W))$
First we find the expectation:
$E(W) = E(\frac{(\bar X)} { 2 } - \frac{(\bar Y)} { 2 }$) = $\frac{1} {2} E(\bar X) - \frac{1} {2} E(\bar Z)$ = $ \frac{ 1 } { 2 } (1) $ - $ \frac{ 1 } { 2 } (0) $ = $ \frac{ 1 } { 2 } $
Then the variance:
$V(W) = V(\frac{(\bar X)} { 2 } - \frac{(\bar Y)} { 2 }$) = $\frac{1} {4} V(\bar X) + \frac{1} {4} V(\bar Z)$ = $ \frac{ 1 } { 2 } (\frac{ 1 } { 2 }) $ + $ \frac{ 1 } { 2 } (\frac{ 1 } { 2 }) $ = $ \frac{ 1 } { 4 } $
In the last sentence, why are $V(\bar X)$ and $V(\bar Z)$ replaced by $\frac{1} {2}$ instead of by 1?
$$V(\bar{X})=V\left(\frac{X_1+X_2}2 \right)=\frac14(V(X_1)+V(X_2))=\frac{1+1}{4}=\frac12$$