Let $X,Y$ be independent random variables with distributions $N(6,7), N(4,5)$ respectively. Let $Z:=3X+5Y$, then:
$1.$ random variables $X, Z$ are independent
$2.\ Cov(X,Z)=24$
$3.\ VarZ=188$
I have to prove or disprove these.
I know that $X\sim N(6,7)$, $Y\sim N(4,5)$ and $X+Y\sim N(10,12)$, so is it true that $3X\sim N(18,21)$, $5Y\sim N(20,25)$ and then $3X+5Y\sim N(38,46)$? If so what should I do next and how can I calculate EZ? Lets assume, I know that when $X\sim N(m,\sigma^2),$ then $EX=m$ and $VarX=\sigma^2$.