prove jointly gaussians of non linear function of gaussian

Question

Let $x_1 , x_2 , x_3 ~ N(0,1)$ iid. $Y = \frac{x_1 + x_2 * x_3}{\sqrt{1+ x_3^2}}$

Does Y and $X_3$ jointly gaussian ?

I already showed that $E[Y|X3]=0 $ and $Var[Y|X3]=0$ And Y|X3 is gaussian... But how can I proceed?

Maybe there is a mistake in the question that make it better?

Answer 1

$ Z=aX_1+bX_2\sim N(0,a^2+b^2).$ Therefore $Y|X_3\sim N(0,1)$ is independent of $X_3$, and $Y, X_3$ are Gaussian and independent.

Let $U,V$ be two rv with joint distribution $\pi(du)K(u,dv)$ where $U\sim \pi$ and $V|U\sim K(u,dv).$ Then $U$ and $V$ are independent if and only if $U$ and $V|U$ are independent.

Proof. $\Rightarrow$ Let $V\sim K(dv)$. We get $\pi(du)K(u,dv)=\pi(du)K(dv)$ and $K(u,dv)=K(dv),$ at least $\pi(du)$ almost everywhere.

$\Leftarrow$ $u\mapsto K(u,dv)$ is a constant.

NN2: True, up to the subtilities of conditioning, I do not quite understand your question.

Answer 2

It seems to me obvious. What about the formal following proof $$E(e^{sY+tX_3})=E[E(e^{sY+tX_3}|X_3)]=E[E(e^{sY}|X_3)\times e^{tX_3}]=e^{s^2/2}\times e^{t^2/2}?$$