Covariance of product A question has been asked regarding the  Variance of product of dependent variables. I am interested in the case in which $X$ is a vector and $Y$ is a scalar and the two variables are independent.
What is $\mbox{var}(\mathbf{X}Y)$, where $\mathbf{X}\perp Y$?
 A: As @whuber pointed out, we can calculate the variance of each entry of $\mathbf{X}Y$ using the formula in the question linked to in my first statement. Thus, $$\mbox{cov}(X_iY, X_iY) = \mbox{var}(X_iY) = \mbox{var}(X_i)\mbox{var}(Y) + \mbox{var}(X_i)\mathbb{E}[Y]^2 + \mbox{var}(Y)\mathbb{E}[X_i]^2$$
Moreover, the covariance between $X_iY$ and $X_jY$ is given by:
$$\begin{align*}
\mbox{cov}(X_iY, X_jY) &= \mathbb{E}[X_iX_jY^2] - \mathbb{E}[X_iY]\mathbb{E}[X_jY]\\
&= \mathbb{E}[X_iX_j]\mathbb{E}[Y^2] - \mathbb{E}[X_i]\mathbb{E}[X_j]\mathbb{E}[Y]\mathbb{E}[Y]\\
&= (\mbox{cov}(X_i, X_j) + \mathbb{E}[X_i]\mathbb{E}[X_j])\mathbb{E}[Y^2] -\mathbb{E}[X_i]\mathbb{E}[X_j]\mathbb{E}[Y]^2\\
&= \mbox{cov}(X_i, X_j)\mathbb{E}[Y^2] + (\mathbb{E}[X_i]\mathbb{E}[X_j])(\mathbb{E}[Y^2] - \mathbb{E}[Y]^2)\\
&= \mbox{cov}(X_i, X_j)\mathbb{E}[Y^2] + \mathbb{E}[X_i]\mathbb{E}[X_j]\mbox{var}(Y)\\
&= \mbox{cov}(X_i, X_j)(\mbox{var}(Y) + \mathbb{E}[Y]^2) + \mathbb{E}[X_i]\mathbb{E}[X_j]\mbox{var}(Y)
\end{align*}$$
Note that when we replace $X_jY$ with $X_iY$ in the formula for $\mbox{cov}(X_iY, X_jY)$, we recover the formula for $\mbox{var}(X_iY)$. Thus, we can construct a covariance matrix $\mathbf{C}$ for $\mathbf{X}Y$ where $C_{ij} = \mbox{cov}(X_iY, X_jY)$.
