Question

Z(u) is a random variable. Z(u) = 3X(u) + 2Y(u) + 2. X(u) and Y(u) are uncorrelated random variables. E[X]=2, VAR[X]=1, E[Y]=1, VAR[Y]=4. Find COV[Z,Y].

Answer 1

The answer key says "Note that the cross-term for variance is zero because X(u) and Y (u) are uncorrelated." but I don't understand why this is the case. Could someone please explain and show work if necessary?

Answer 2

\[\newcommand{cov}{\mathbin{\text{Covar}}}
\begin{align*}
\cov(Z,Y)&=\mathbb{E}\left[\bigg(Z-\mathbb{E}(Z)\bigg)\bigg(Y-\mathbb{E}(Y)\bigg)\right]\\\\
&=\mathbb{E}(ZY)-\mathbb{E}(Z)\mathbb{E}(Y)\\\\
&=\mathbb{E}\bigg((3X+2Y+2)Y\bigg)-\mathbb{E}(3X+2Y+2)\mathbb{E}(Y)\\\\
&=3\color{red}{\mathbb{E}(XY)}+2\color{blue}{\mathbb{E}(Y^2)}+2\mathbb{E}(Y)-3\color{red}{\mathbb{E}(X)\mathbb{E}(Y)}-2\color{blue}{\bigg(\mathbb{E}(Y)\bigg)^2}\\
&\quad\quad -2\mathbb{E}(Y)\\\\
&=3\color{red}{\cov(X,Y)}+2\color{blue}{\mathbb{V}(Y)}
\end{align*}\]
and the red term is zero because we know \(X\) and \(Y\) are uncorrelated.