$\xi_{1}, \ldots, \xi_{5}$ and independent random variables with $N_{1,4}$ distribution. Find distribution of
$\eta = \xi_{1}-\frac{\xi_{2}+\xi_{3}+\xi_{4}+\xi_{5}}{2}$
and its CDF.
My solution
We know that if $X \in N_{a,\alpha^2}$ and $Y \in N_{b,\beta^2}$ then $X+Y=Z \in N_{a+b,\alpha^2 + \beta^2}$
Also, $X=cZ \in N_{a,(c\alpha)^2}$, where $c =const$
So,
$\eta \in N_{1-\left(\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}\right), 4-(1+1+1+1)} = N_{0,0}$
Basically, I get Dirac delta function which is weird (or not?). Is it correct?
If $X$ and $Y$ are independent random variables then $var (X-Y)=var (X)+var (Y)$, not $var (X-Y)=var (X)-var (Y)$.