I need to create a covariance matrix of two random variables where $W= \text{the wingspan of a swallow}$ and $V= \text{velocity}$.
$W$ has a Normal distribution with mean $10$ and standard deviation $4$.
$V=0.5W+U$ where $U$ is a random variable for error. $U$ has a standard Normal distribution (mean $0$, standard deviation $1$) and is independent of $W$.
Since the distributions we are working with are Normal, we have $V(W)=4^2=16$ and we have $V(V)=V(0.5W+U)=(0.5)^2V(W)+V(U)=(0.25)(16)+(1)=5$.
For the covariance, I believe I should be using the equation $V(X+Y)=V(X)+V(Y)+2\mathrm{Cov}(X,Y)$. However, I don't know what $V(X+Y)$ equals and thus can't solve for $\mathrm{Cov}(X,Y)$. Is there something that I am missing here?
$$\mathrm{Cov}(W,V)=E[(W-E[W])(V-E[V])] \\ =E[(W-10)(V-5)] \\ =E[(W-10)(0.5W+U-5)].$$
Now you see that you need $E[W],E[U],E[W^2]$ and $E[WU]$. It's in the last one that the independence comes into play.