A problem on bivariate random variables

Question

Suppose we have absolutely continuous random vectors $X=(X_1,X_2)$ and $Y=(Y_1,Y_2)$. And we have $Y_i=a_iX_i+b_i$, and $a_i>0, b_i\geq 0$ $i=1,2$ . Let ${F}$ be a distribution function such that ${F}_X(x_1,x_2)=P(X_1\leq x_1,X_2\leq x_2).$ We know that $\frac{\partial^2}{\partial x_1\partial x_2}F_X(x_1,x_2)=f_X(x_1,x_2).$

Now, \begin{align} F_Y(y_1,y_2)&=P(Y_1\leq y_1,Y_2\leq y_2)\\ &=P(a_1X_1+b_1\leq y_1,a_2X_2+b_2\leq y_2)\\ &=P(X_1\leq (y_1-b_1)/a_1,X_2\leq (y_2-b_2)/a_2)\\ &=F_X((y_1-b_1)/a_1,(y_2-b_2)/a_2)\\ &=F_X(x_1,x_2) \end{align} Form here can we say that $f(y_1,y_2)=f(x_1,x_2)$? Thanks beforehand!

Answer 1

Remember, when $\mathbf X\mapsto \mathbf Y$ via a one-one onto transformation, say $y_i:= g_i(x_1, \ldots, x_n), $ and $h_i:=g_i^{-1}, $ then if $\frac{\partial h_i}{\partial y_j}$ are continuous for all $i, j\in\{1, 2,\ldots, n\}, $ then $$f_\mathbf Y(\mathbf y) =f_\mathbf X(h_1(\mathbf y), \ldots, h_n(\mathbf y) ) |\mathrm J(\mathbf y) |, $$ where $\mathrm J(\mathbf y) :=\det\left(\frac{\partial h_i}{\partial y_j}\right) . $