# Rejection sampling from a Gamma distribution using a Cauchy proposal

i'm trying to find the parameters $\gamma,x_0$ of a standard Cauchy distribution : $$T(x)= \frac{1}{(\pi \gamma (1+(\frac{x-x_0}{\gamma})^2))}$$ To perform rejection sampling from a gamma distribution: $$G(x) = \frac{1}{\Gamma(\alpha)}x^{\alpha-1}\lambda^\alpha \exp(-\lambda x)$$ And i have seen than in several textbooks as Bishop's pattern recognition it seems like if is straightforward to derive those parameters to obtain the values $$x_0 = \frac{\alpha -1}{\lambda}$$ and $$\gamma^2 = 2\alpha-1$$ But so far i just have obtained $x_0$ by making equal the derivatives of T and G and solving for x, but for the $\gamma$ parameter i just dont know how to find it, because all the attempts that i have made leads me to polynomial equations of degree >=3 and irrational roots... Thanks for your help!

• Common usage would have lower case for density functions and upper case for cdfs. Why are you setting derivatives to be equal? Is there some lower bound on $\alpha$? – Glen_b Mar 13 '14 at 4:25
• (I can see a reason to set the derivatives equal, but if you're doing it for a different reason, that would be important to know). Oh, and is this for some subject? – Glen_b Mar 13 '14 at 4:30
• Hi @Glen_b thanks for the clarification for notation, there are two main reasons to make equal these derivatives: 1) Setting the derivatives to be equal to see where they modes should be centered 2) Im not pretty sure of this ... but maximizing the proposal will maximize the reason $G/(mT)$. – sebastianffx Mar 13 '14 at 5:19
• And yeap, this is for an independent class on machine learning from a probabilistic perspective, following the Murphy's book of the same name, i have discussed this exercise with some people but we still trying to solve it. – sebastianffx Mar 13 '14 at 5:34
• This would fall under self-study. Please add the tag and read the tag wiki info. You'll need to lose one of your present tags to do so. I think sampling is redundant (given you have rejection-sampling), so I'd suggest replacing that one. If you follow those guidelines, you'll likely get some hints/guidance. – Glen_b Mar 13 '14 at 6:05