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I am trying to do this using the Poisson-Gamma mixture parameterization.

I think the derivation is related to the derivation of the expected value. But I have only seen how to derive the mean using conditioning, not from summation. Could someone help?

kjetil b halvorsen
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Ka Lee
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  • Is your question that you're trying to prove that the neg binomial is a mixture of poison and Gamma? – AdamO Jan 09 '23 at 18:37
  • No, I am trying to understand when negative binomial is set up as a mixture of Poisson/Gamma, how to derive its Laplace Transform? – Ka Lee Jan 09 '23 at 19:38
  • See https://stats.stackexchange.com/questions/285378/moment-generating-function-negative-binomial, https://stats.stackexchange.com/questions/236728/negative-binomial-mgf-converges-to-poisson-mgf – kjetil b halvorsen Jan 09 '23 at 19:41

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