Find unconditional distribution using transforms

[psie]

Homework Statement

If $X\mid \Sigma^2=\lambda\in N(0,1/\lambda)$ with $\Sigma ^2\in \Gamma \left(\frac{n}{2}{,}\frac{2}{n}\right)$, show that $X\in t(n)$ using transforms.

Relevant Equations

$t(n)$ is the t-distribution with parameter $n$.

I am asked to solve the challenging problem above (I don't see the purpose in this exercise actually, since transforms just make it harder I think).

Here's my attempt; denote by $\varphi_X$ the characteristic function (cf) of $X$, then $$\varphi_X(t)=Ee^{itX}=E(E(e^{itX}\mid\Sigma^2))=Eh(\Sigma^2),$$where $h(\lambda)=\varphi_{X\mid \Sigma^2=\lambda}(t)=e^{-t^2/(2\lambda)}$, since recall the cf of $N(0,\sigma^2)$ is just $e^{-t^2\sigma^2/2}$. Now, $$\varphi_X(t)=Ee^{-t^2/(2\Sigma^2)}=\ldots,$$and I don't know how to proceed further. The thing is, I prefer not to take this route, since the cf of $t(n)$ is very intricate, but I guess this is the way to go, by comparing cfs, or?

 

[psie]

It's possible to solve this by comparing cfs, but it is a bit of work. What will come in handy are equations 10.32.10 and 10.32.11 here. We have $$\varphi_X(t)=E\left[e^{-t^2/(2\Sigma^2)}\right]=\frac{\left(n/2\right)^{n/2}}{\Gamma(n/2)}\int_0^\infty \exp\left(-\frac{n\lambda}{2}-\frac{t^2}{2\lambda}\right)\lambda^{n/2-1}\,d\lambda.$$ Now make a substitution so as to identify the integral representation of equation 10.32.10 given in the link. To derive the cf of the Student's $t$-distribution with parameter $n$, you'll need equation 10.32.11. Equation 10.32.9 in that link shows that $K_\nu(z)=K_{-\nu}(z)$, which is also useful.