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I have a Bayesian stat question where I am a little confused.

Let $N_{1}$ and $N_{2}$ independent having $\mbox{Poisson}(\Lambda)$ distribution and $\Lambda \sim \mbox{Gamma}(\alpha,\theta)$. Then how we prove that $N_{1} + N_{2} \sim \mbox{Poisson}(2 \Lambda)$ because it could be the case that $N_{1} \sim \mbox{Poisson}(2)$ and $N_{2} \sim \mbox{Poisson}(17)$?

Thank you!

Matt
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  • It depends on whether "Let $N_{1}$ and $N_{2}$ independent having $\mbox{Poisson}(\Lambda)$ distribution" means they share the same $\Lambda$, or whether $N_1 \sim \mbox{Poisson}(\Lambda_1)$ and $N_2 \sim \mbox{Poisson}(\Lambda_2)$ – Henry Feb 17 '22 at 14:23

1 Answers1

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To avoid ambiguity, it would be preferable to write the given as

$$N_i|\Lambda=\lambda \quad \overset{iid}{\sim } Poiss(\lambda),$$

and the claim you are trying to show as

$$N_1+N_2|\Lambda=\lambda \quad \overset{iid}{\sim } Poiss(2\lambda).$$

Then the claim is easily shown using characteristic functions:

$$\varphi_{N_1+N_2|\Lambda =\lambda}(t)=\varphi_{N_1|\Lambda =\lambda}(t)\varphi_{N_2|\Lambda =\lambda}(t)=(\exp(\lambda (e^{it}-1)))^2=\exp(2\lambda (e^{it}-1)).$$

The distribution of $\Lambda$ is irrelevant.

Golden_Ratio
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  • Thank you for an answer and I am familiar to show that $N_{1}+N_{2} \sim \mbox{Poisson}(2\lambda)$ given that $N_{i} , | , \lambda \stackrel{iid}{\sim} \mbox{Poisson}(\lambda)$. But the situation I am asking here is that $N_{1}$ and $N_{2}$ are simply independent but not identically distributed. – Matt Feb 17 '22 at 15:20
  • @Matt So you mean you are given $N_i|\Lambda_i=\lambda_i\sim Poiss(\lambda_i)$? But then what precisely is the claim that you are asked to show in that case? – Golden_Ratio Feb 17 '22 at 15:24
  • The problem is this, let $N_{i} , | , \lambda \sim \mbox{Poisson}(\lambda)$ for $i = 1,2$ where $i=1$ represents claim couts for a person and $i=2$ represents the claim counts for a different and independent person. Clearly $\lambda$ varies from person to person but what happens to $N_{1} + N_{2}$? – Matt Feb 17 '22 at 15:28