A normal scale mixture with a $\chi^2_1$ precision: $X$ is Cauchy

Let X and Y be two random variables defined on the same probability space such that for each $y>0$, the conditional density function of X given $Y=y$ is $f(x\mid y)=\sqrt{\dfrac y{2\pi}}\,e^{-yx^2/2}$, $y>0$, $x\in\mathbb{R}$. Let $g(y)=\dfrac1{\sqrt{2\pi y}}e^{-y/2}$, $y>0$. Which of the following statements is more appropriate?