Poisson distribution and idd random variables - Proof of an equality

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I want to solve the following task

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The first part was very easy for me.

But I dont know how to solve the second one. I guess I even understood it completely.

I am thankful for any kind of help!

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Hints:

  • By exchangeability, $E(Sg(S)\mid N)=Nh(N)$ with $h(n)=E(X_1g(X_1+\cdots+X_n))$
  • For every $n$, $h(n+1)=E(X_0g(X_0+X_1+\cdots+X_n))$
  • Apply the first part of the exercise to $h$