Consider two random variables $X$ and $Y$ such that X ∼ U(0, 1) and $Y|X=x$ ∼ N($x$, $x^2$), where $U$ and $N$ denotes an Uniform and Normal distributions, respectively. Prove that $Y/X$ and $X$ are independent.
I tried using the variable change method, but $X,Y$ are not independent (cov($X,Y$) is not $0$). As they are independent, I can't simply multiply their pdf's to get the joint pdf. So, I can't proceed further using this method. Can anyone help me in proving this?
I think the best way to prove this is through characteristic functions. We have $E(e^{itY}|X)=e^{itX}e^{-t^{2}X^{2}/2}$. Replace $t$ by $\frac t X$ to get $E(e^{it\frac Y X}|X)=e^{it}e^{-t^{2}/2}$. Now consider $Ee^{it\frac Y X +isX}$. We have $E(e^{it\frac Y X +isX}|X)=e^{it}e^{-t^{2}/2}e^{isX}$. Take expectation to get $Ee^{it\frac Y X +isX}=e^{it}e^{-t^{2}/2}Ee^{isX}$. The fact that the joint characteristic function of $\frac Y X$ and $X$ at $(t,s)$ is the product of a function of $t$ and a function of $s$ implies that these two variables are independent. Note that the distribution of $X$ plays no role.