Problem expressing full conditionals I have this problem,
$Y_{i}$~Gamma($\alpha$,$\beta$) 1...N 
$\alpha$~Exp($\lambda$) 
$\beta$~Exp($\lambda$)
$\lambda$=0.001,Find the full conditionals.
I have done the following: 
p($\theta$|Y)$\propto$p(Y|$\theta$)p($\theta$) = $\prod_{i}^N Y_{i}^{\alpha-1} e^{-\beta Y_{i}} \lambda e^{\lambda \alpha} \lambda e^{-\lambda \beta}$ 
= $\prod_{i}^N Y_{i}^{\alpha-1} e^{-\beta Y_{i}} \lambda^{2} e^{-(\lambda \beta+\lambda \alpha)}$ 
= $\prod_{i}^N \lambda^{2} Y_{i}^{\alpha-1} e^{-(\lambda \beta+\lambda \alpha+\beta Y_{i})}$ 
Then 
p($\alpha$|$\beta ,Y)$$\propto Y_{i}^{\alpha-1}e^{-\lambda \alpha}$ 
p($\beta$|$\alpha ,Y)\propto$$ e^{-(\lambda + \sum_{1}^{N}{Y_{i}})\beta}$ 
Now these look like a gamma and exponential, but I'm having problems adding the normalizing constants or expressing them as the functions, Gamma($\alpha, \lambda \alpha$) and Exp($\lambda + \sum_{1}^{N}{Y_{i}}$)?. Any help? 
 A: The joint distribution of "everything" $(y_1,\ldots,y_N,\alpha,\beta)$ is
$$
\prod_{i}^N \left\{ \dfrac{\beta^\alpha y_{i}^{\alpha-1}}{\Gamma(\alpha)} \right\} e^{-\beta y_{i}} \lambda^{2} e^{-(\lambda \beta+\lambda \alpha)}\,,
$$
incorporating all "constants". Therefore, if we only take the terms that involve $\alpha$ in the above we find:
$$
\beta^{N\alpha}  e^{-\lambda \alpha} \Gamma(\alpha)^{-N} \prod_{i}^N y_{i}^{\alpha-1} = \Gamma(\alpha)^{-N} \exp\left\{ N\alpha\log(\beta) +\sum_{i=1}^N (\alpha-1)\log(y_i) -\lambda\alpha \right\}
$$
meaning that
$$
p(\alpha|y_1,\ldots,y_N,\beta,\lambda) \propto
\Gamma(\alpha)^{-N} \exp\left\{ -\alpha \left[\lambda-N\log(\beta)-\sum_{i=1}^N \log(y_i) \right] \right\}
$$
which is not a standard distribution...
Similarly, if we extract all terms that depend on $\beta$ from the joint, we obtain
$$
\beta^{N\alpha} e^{-\lambda \beta} \prod_{i}^N e^{-\beta y_{i}}
$$
that is,
$$
p(\beta|y_1,\ldots,y_N,\alpha,\lambda) \propto
\beta^{N\alpha} \exp\left\{ -\beta\left[\lambda +\sum_{i=1}^N y_i \right] \right\}\,.
$$
As a function of $\beta$, this is proportional to the Gamma density $$\mathcal{G}a\left(N\alpha+1,\lambda +\sum_{i=1}^N y_i\right)$$ and hence this Gamma is the full (conditional) posterior distribution of $\beta$.
