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How to show the following:

Using the student t-distribution with $k > 0$ degrees of freedom, location parameter $l$ and scale parameter $s$ having density

$ \frac{\Gamma ((k+1)/2)}{\Gamma ((k/2) \sqrt{k \pi s^2}} \{ 1 + k^{-1}( \frac{x-l}{s} )\}^{-(k+1)/2}$

How to show that $t$-distribution can be written as a mixture of Gaussian distributions by by letting $X$~ $N(\mu,\sigma^2)$, $\tau = 1/\sigma^2$~$\Gamma(\alpha,\beta)$ and integrating the joint density $f(x,\tau|\mu)$ to get marginal density $f(x|\mu)$. What are the parameters of the resulting $t$-distribution, as functions of $\mu,\alpha,\beta$

I got lost in calculus. By integrating the joint conditional density with the Gamma distribution. Please help!

Thanks a lot!

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2 Answers 2

up vote 8 down vote accepted

The PDF of a Normal distribution is

$$f_{\mu, \sigma}(x) = \frac{1}{\sqrt{2 \pi} \sigma} e^{-\frac{(x-\mu )^2}{2 \sigma ^2}}dx$$

but in terms of $\tau = 1/\sigma^2$ it is

$$g_{\mu, \tau}(x) = \frac{\sqrt{\tau}}{\sqrt{2 \pi}} e^{-\frac{\tau(x-\mu )^2}{2 }}dx.$$

The PDF of a Gamma distribution is

$$h_{\alpha, \beta}(\tau) = \frac{1}{\Gamma(\alpha)}e^{-\frac{\tau}{\beta }} \tau^{-1+\alpha } \beta ^{-\alpha }d\tau.$$

Their product, slightly simplified with easy algebra, is therefore

$$f_{\mu, \alpha, \beta}(x,\tau) =\frac{1}{\beta^\alpha\Gamma(\alpha)\sqrt{2 \pi}} e^{-\tau\left(\frac{(x-\mu )^2}{2 } + \frac{1}{\beta}\right)} \tau^{-1/2+\alpha}d\tau dx.$$

Its inner part evidently has the form $\exp(-\text{constant}_1 \times \tau) \times \tau^{\text{constant}_2}d\tau$, making it a multiple of a Gamma function when integrated over the full range $\tau=0$ to $\tau=\infty$. That integral therefore is immediate (obtained by knowing the integral of a Gamma distribution is unity), giving the marginal distribution

$$f_{\mu, \alpha, \beta}(x) = \frac{\sqrt{\beta } \Gamma \left(\alpha +\frac{1}{2}\right) }{\sqrt{2\pi } \Gamma (\alpha )}\frac{1}{\left(\frac{\beta}{2} (x-\mu )^2+1\right)^{\alpha +\frac{1}{2}}}.$$

Trying to match the pattern provided for the $t$ distribution shows there is an error in the question: the PDF for the Student t distribution actually is proportional to

$$\frac{1}{\sqrt{k} s }\left(\frac{1}{1+k^{-1}\left(\frac{x-l}{s}\right)^2}\right)^{\frac{k+1}{2}}$$

(the power of $(x-l)/s$ is $2$, not $1$). Matching the terms indicates $k = 2 \alpha$, $l=\mu$, and $s = 1/\sqrt{\alpha\beta}$.

Notice that no Calculus was needed for this derivation: everything was a matter of looking up the formulas of the Normal and Gamma PDFs, carrying out some trivial algebraic manipulations involving products and powers, and matching patterns in algebraic expressions (in that order).

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Inspired by this answer I made an animation of the t distribution as a mixture of normal distributions. It is available here: sumsar.net/blog/2013/12/t-as-a-mixture-of-normals –  Rasmus Bååth Dec 6 '13 at 20:42

I don't know the steps of the calculation, but I do know the results from some book (cannot remember which one...). I usually keep it in mind directly... :-) The Student $t$ distribution with $k$ degree freedom can be regarded as a Normal distribution with variance mixture $Y$, where $Y$ follows inverse gamma distribution. More precisely, $X$~$t(k)$,$X$=$\sqrt Y$*$\Phi$,where $Y$~$IG(k/2,k/2)$,$\Phi$ is standard normal rv. I hope this could help you in some sense.

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