Distribution of transformed random variables

Question

We have that f is a density w.r.t the lebesgue measure $m$ for a probability measure on $\mathbb{R}$, that f is continuous and strictly positive. X and Y are to random variables s.t. the distribution of (X,Y) has density $g(x,y)=f(x)f(y)$ w.r.t. $m_2$. Furthermore we define

$ \begin{pmatrix} \tilde{X} \\ \tilde{Y} \end{pmatrix}=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} X \\ Y \end{pmatrix} $

for $a,b,c,d \in \mathbb{R}$, i.e. $\tilde{X}=aX+bY$ and $\tilde{Y}=cX+dY$. We know that both $X$ and Y, and $\tilde{X}$ and $\tilde{Y}$ are uncorrelated.

Now I have to find an expression $\tilde{g}$ for the density of $\tilde{X},\tilde{Y}$ w.r.t. $m_2$ when $ad-bc\neq 0$.

I cannot find out which formula to use to find this expression, can anyone help?

Answer 1

Hint: As long as the determinant of the matrix is non zero, the mapping between $(X,Y)$ and $(\tilde X,\tilde Y)$ is bijective, so $\tilde g(\tilde x, \tilde y) = g(x,y)|J(\tilde x,\tilde y)|$, where:

$$\begin{align} \begin{pmatrix} x \\ y\end{pmatrix} & = \begin{pmatrix} a & b\\ c & d\end{pmatrix}^{-1}\begin{pmatrix} \tilde{x} \\ \tilde{y}\end{pmatrix} \\ J(\tilde x,\tilde y) & = \begin{Vmatrix}\frac{\mathrm d x}{\mathrm d \tilde x} & \frac{\mathrm d x}{\mathrm d\tilde y} \\ \frac{\mathrm d y}{\mathrm d\tilde x} & \frac{\mathrm d y}{\mathrm d\tilde y}\end{Vmatrix} \end{align}$$

So just substitute for $x$ and $y$.