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I need help finding a distribution in order to solve an exercise. We have the following problem:

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Following this we want to calculate:

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So I have found a complete sufficient statistic which is $\sum_{i=1}^n (Y_i)$. In order to find the $UMVUE$, I first need to find the distribution of $\sum_{i=1}^n (Y_i)$ | $β$. Can someone tell me how to do this??

I tried doing \begin{align} F_T(t) &= \mathrm P(T\leq t) = \mathrm P(\sum_{i=1}^n (Y_i)\leq t) \\ \end{align} How do I procede from here?

Thanks in advance for the help!

Diogo Fonseca
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  • Hint: The sum of two Poisson distributions is again a Poisson distribution. – Math1000 Nov 19 '19 at 02:44
  • See https://math.stackexchange.com/questions/2689276/finding-an-unbiased-estimator-of-e-2-lambda-for-poisson-distribution?noredirect=1&lq=1, https://math.stackexchange.com/questions/2689415/for-what-values-of-k-in-mathbbz-setminus-0-does-there-exist-an-unbiased?noredirect=1&lq=1, https://stats.stackexchange.com/questions/349942/umvue-of-e-lambda-from-poisson-distribution?noredirect=1&lq=1. An unbiased estimator of $\psi(\beta)$ based on $\sum Y_i$ is the required UMVUE. – StubbornAtom Nov 19 '19 at 12:16

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