Calc-Based Stats, Prove a Conditional Distribution is Gamma Distributed

[aleph_0]

Homework Statement

An image of the assigned problem is here: http://imgur.com/aYkaM

Homework Equations

The formula for being exponential, gamma, and probably Bayes's Law. They'd take a while to type out, and presumably anyone who can help me with this already knows the formulas or will understand them by looking at my solution thus far.

The Attempt at a Solution

$P(X_{1}=x_{1}, ..., X_{n}=x_{n}|W=w)P(W=w) \quad = \\\\ P(W=w|X_{1}=x_{1}, ..., X_{n}=x_{n})P(X_{1}=x_{1}, ..., X_{n}=x_{n}) \quad \Longrightarrow \\\\ P(W=w|X_{1}=x_{1}, ..., X_{n}=x_{n}) \quad = \quad \frac{P(X_{1}=x_{1}, ..., X_{n}=x_{n}|W=w)P(W=w)}{P(X_{1}=x_{1}, ..., X_{n}=x_{n})} \quad = \\\\ \frac{w^{n}e^{-w(x_{1}+...+x_{n})}\frac{\beta^{t}}{\Gamma (t)}w^{t-1}e^{-\beta w}}{P(X_{1}=x_{1}, ..., X_{n}=x_{n})} \quad = \quad \frac{\beta^{t}}{\Gamma (t) P(X_{1}=x_{1}, ..., X_{n}=x_{n})}{w^{t+n-1}e^{-w(\beta + \sum_{i=1}^{n}x_{i})}}$

Now at this point I feel like I've gone most of the way, but the only thing is the "coefficient". For this to be a true gamma distribution with the said parameters, that front factor needs to be $\frac{(\beta+\sum x_{i})^{t+n}}{\Gamma(t+n)}$ and I just can't see how to make that happen.

 

[Ray Vickson]

Homework Statement

An image of the assigned problem is here: http://imgur.com/aYkaM

Homework Equations

The formula for being exponential, gamma, and probably Bayes's Law. They'd take a while to type out, and presumably anyone who can help me with this already knows the formulas or will understand them by looking at my solution thus far.

The Attempt at a Solution

$P(X_{1}=x_{1}, ..., X_{n}=x_{n}|W=w)P(W=w) \quad = \\\\ P(W=w|X_{1}=x_{1}, ..., X_{n}=x_{n})P(X_{1}=x_{1}, ..., X_{n}=x_{n}) \quad \Longrightarrow \\\\ P(W=w|X_{1}=x_{1}, ..., X_{n}=x_{n}) \quad = \quad \frac{P(X_{1}=x_{1}, ..., X_{n}=x_{n}|W=w)P(W=w)}{P(X_{1}=x_{1}, ..., X_{n}=x_{n})} \quad = \\\\ \frac{w^{n}e^{-w(x_{1}+...+x_{n})}\frac{\beta^{t}}{\Gamma (t)}w^{t-1}e^{-\beta w}}{P(X_{1}=x_{1}, ..., X_{n}=x_{n})} \quad = \quad \frac{\beta^{t}}{\Gamma (t) P(X_{1}=x_{1}, ..., X_{n}=x_{n})}{w^{t+n-1}e^{-w(\beta + \sum_{i=1}^{n}x_{i})}}$

Now at this point I feel like I've gone most of the way, but the only thing is the "coefficient". For this to be a true gamma distribution with the said parameters, that front factor needs to be $\frac{(\beta+\sum x_{i})^{t+n}}{\Gamma(t+n)}$ and I just can't see how to make that happen.

$$P(X_1=x_1,\ldots,X_n=x_n) = \int P(X_1=x_1,\ldots,X_n=x_n|w) f_W(w) \, dw.$$ Just do the integral.

RGV