I have been working with this question and can't really figure it out!
Let $Y$ be Cauchy distributed and let $X$ be $\chi^{2}$ distrubted with 1 degree of freedom
Show that $E(XY^{2}) = 1$
I have tried to show that X and $Y^{2}$ is independent in order to find the density function for the vector $(X,Y^{2})$ but with no luck!
additional information:
The function $p(x,y) = \dfrac{e^{-\frac{x}{2}(1+y^{2})}}{2\pi}$ for $x>0, y\in \mathbb{R}$
is a densitiy function on $(0,\infty )$x$\mathbb{R}$
Let $(X,Y)$ be a continuous random variable with densitiy p