# The distribution of the product of a multivariate normal and a lognormal distribution

If $$X=\left(\begin{array}{c} X_{1}\\ X_{2} \end{array}\right)\sim N\left[\left(\begin{array}{c} \mu_{X_{1}}\\ \mu_{X_{2}} \end{array}\right),\left(\begin{array}{cc} \sigma_{X_{1}}\\ \sigma_{X_{1}X_{2}} & \sigma_{X_{2}} \end{array}\right)\right]$$ and $$Y\sim logN\left(\mu_{Y},\sigma_{Y}\right)$$ then, which is the distribution of $YX=\left(\begin{array}{c} YX_{1}\\ YX_{2} \end{array}\right)$? is it lognormal? which are the mean and variance-covariance matrix of YX? We assume that $X$ and $Y$ are independent.

I will start looking at some simpler case. Let $X, Z$ be independent standard normal random variables, and let $Y=e^Z$. Then $Y$ is lognormal, and we are interested in the distribution of the product $XY= Y \mid X \mid S$ where $S=\text{sign}(X)$ is independent of $X$. I will study the distribution of $\mid XY\mid = \mid X \mid Y$. Then finally multiplying by $S$ simply represents a random reflection in the vertical axis. Using the Mellin transform simplifies working with the product of independent (and positive) random variables. The Mellin transform of (almost surely) positive $X$ is $$\DeclareMathOperator{\E}{\mathbb{E}} M_X(s) = \E X^s = \E e^{\log(X^s)}=\E e^{s \log X} =K_{\log X}(s)$$ where $K$ represents the moment generating function (mgf). So the Mellin transform of $X$ is simply the mgf of $\log X$.
For our random variables we find (I used maple here) $$M_{\mid X \mid}(s) = \frac{2^{s/2}\Gamma((s+1)/2)}{\sqrt\pi} \\ M_Y(s) = e^{s^2/2}$$ Then for the Mellin transform of the product: $$M_{\mid X\mid Y}(s) =\E (\mid X \mid Y)^s=\E (\mid X\mid^s Y^s)=\\ \E \mid X \mid^s \E Y^s \qquad \text{(by independence)} \\ = \frac{e^{(s\log 2 + s^2)/2}\Gamma((s+1)/2)}{\sqrt\pi}$$ Then we need the inverse transform of this, or we can use it as a basis for approximation, maybe saddlepoint approximation.