Consider: $$ X(t)=F^{-1} \{\Phi[Y(t)]\} $$ Where: F is any distribution function. Y is a stationary Gaussian process with zero mean and unit variance, $\Phi$ is the Gaussian distribution function.
Show that the second density function, $$f(x_1,x_2;t_1,t_2)=\frac{d^2F(x)}{dx_1dx_2} $$
Takes the form
$$\frac{f_1(x_1)f_1(x_2)}{\sqrt{1-\rho^2}}exp\{-\frac{\rho}{2(1-\rho^2)}[\rho(y_1^2+y_2^2) -2y_1y_2]$$
Where: $$f_1(x)=\frac{dF(x)}{dx}$$ $$\rho=E[Y(t_1)Y(t_2)]$$ $$y=\Phi^{-1}[F(x)]$$
Clearly I have everything I need to do this problem, and I'm just having the mother of all brain-farts and can't piece it together. I imagine something with the chain rule: e.g. $\frac{dF}{dx}=\frac{dF}{dy}\frac{dy}{dx}$. I find $\frac{dy}{dx}=\frac{1}{\phi(\Phi^{-1}(F(x))}f_1(x)$ So, I feel, in the end, I should get something along the lines of: $$\frac{1}{\phi(\Phi^{-1}(F(x_1))}\frac{1}{\phi(\Phi^{-1}(F(x_2))}f_1(x_1)f_1(x_2)\phi(y_1,y_1)$$ Where $\phi$ is the normal pdf (bivariate normal in the case of y1 and y2). This is similar to what I'm supposed to find, but, there are those pesky fractions in front that I don't what I'm supposed to do with. What am I missing? Any help would be greatly appreciated.