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Let $Y$ be the sum of $n$ independent observations frm a $pois(\theta)$ distribution.

Further let the prior distribution for $(\theta)$ be $\gamma(\alpha,\beta)$.

I need to find the posterior distribution of $\theta$, given that $Y=y$

AND find a point estimate of $\theta$ given this value of y.

Can someone please help me to do this?

Completely stuck on how i would do this, but im guessing that a point estimate is the mean of the distribution, how would i actually find this?

user67411
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1 Answers1

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By definition, $$P[Y=y,\theta\in\mathrm dx]=\mathrm e^{-nx}\frac{(nx)^y}{y!}\beta^{\alpha}\Gamma(\alpha)^{-1}x^{\alpha-1}\mathrm e^{-\beta x}\mathbf 1_{x\gt0}\mathrm dx. $$ Hence, $$ P[\theta\in\mathrm dx\mid Y=y]\propto\mathrm e^{-(n+\beta)x}x^{y+\alpha-1}\mathbf 1_{x\gt0}\mathrm dx. $$ Thus, conditionally on $Y=y$, the distribution of $\theta$ is $\gamma(\alpha+y,\beta+n)$. In particular, $$ E[\theta\mid Y=y]=\frac{\alpha+y}{\beta+n}. $$

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