" Consider 2 random variables $X_1$ and $X_2$:
- $X_1 = Y_1 \sqrt{\alpha + \beta {X_0}^2}$
- $X_2 = Y_2 \sqrt{\alpha + \beta {X_1}^2}$
$ \alpha>0 $ , $\beta \ge 0$
where $Y_1$ and $Y_2$ are independent and both follow a normal standard distribution $N (0, 1)$. Also, $Y_1$ is independent of $X_0$ and $Y_2$ is independent of $X_1$.
Find the conditional probability distributions of $X_1|X_0 = x_0$ and $X_2|X_1 = x_1$. "
I'm guessing it has to do with the normal joint density but I don't know where to start since no information was given about $X_0$. Is it possible that $f(y_1) = f(x_1,x_0)$ and $f(y_2) = f(x_1,x_2)$ which would led to a bivariate normal joint density?
Thank you in advance for your help!