= Normal scale mixture representation of Student t
{title2=$T=Z/\sqrt{\Lambda/\nu},\quad\Lambda\sim\chi_\nu^2$}
For <independent> $Z\sim N(0,1)$ and $\Lambda\sim\chi_\nu^2$, the displayed ratio has the <Student t-distribution> with $\nu>0$ <statistical degrees of freedom>. Equivalently, conditional on $\Lambda=\lambda>0$, $T$ has <normal distribution> $N(0,\nu/\lambda)$. This is a <normal scale mixture representation of Student t>: a small latent denominator occasionally produces a much larger spread. The result is a <heavy-tailed distribution> even though every conditional <probability distribution> is a <normal distribution>.
For a short derivation, <independence> lets us multiply the conditional <normal density> and the <chi-squared distribution>'s <probability density function>, then integrate over the unobserved $\lambda$:
$$
f_\nu(t)=\int_0^\infty \sqrt{\frac{\lambda}{2\pi\nu}}e^{-\lambda t^2/(2\nu)}\frac{\lambda^{\nu/2-1}e^{-\lambda/2}}{2^{\nu/2}\Gamma(\nu/2)}\,d\lambda.
$$
The resulting <gamma function> integral has shape $(\nu+1)/2$ and rate $(1+t^2/\nu)/2$, yielding the normalized <probability density function> above. Also, for $\nu>2$, <conditional expectation> gives $\mathbb E[T^2]=\nu\mathbb E[\Lambda^{-1}]=\nu/(\nu-2)$. Thus unit scale does not mean unit <variance>. Finally, $\mathbb E[\Lambda/\nu]=1$ and $\operatorname{Var}(\Lambda/\nu)=2/\nu$ imply <convergence in probability> to 1 by <Chebyshev's inequality>; <Slutsky's theorem> then gives the <standard normal distribution> limit.
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