Jtdylktuy

So I'm trying to derive an expected value (related to Bayesian risk/loss function) and I've derived all except one final part. To finish the final part I need to derive one of the following expected values (either will work)

Define the probability mass function

$$p_N (n) = {n-1 choose x-1} frac{Gamma(a+b)}{Gamma(a) + Gamma(b)}frac{Gamma(a+x)+Gamma(b+n-x)}{Gamma(a+b+n)}$$

for $n =x,x+1,x+2,dots$ and also define the conditional pmf

$$p_N(n|p) = {n-1 choose x-1} p^x (1-p)^{n-x}$$

To complete the final step I need either one of:

$$E_N left[ left(frac{x-1}{N-1} right)^2right]$$

or

$$E_{N} = left[ left(frac{x-1}{N-1} right)^2 bigg| p right]$$

In previous questions, I've derived the necessary expected values by absorbing the terms into the probability functions in order to construct a new distribution function, and then obtaining the expected value by normalizing it so that it sums/integrates to $1$. But for these ones I'm stuck due to the fact it's squared and you're left with a single fraciton that can't be absorbed into the combination.

Does anyone see a way forward?

Hint using generating functions :

begin{align*}
sum_{n=x}^infty {n-1choose x-1} frac{1}{(n-1)^2} q^n&=qsum_{n=x}^infty {n-1choose x-1} frac{1}{(n-1)^2} q^{n-1}\
&=qsum_{n=x}^infty {n-1choose x-1} int frac{1}{(n-1)} q^{n-2} dq\
&=qsum_{n=x}^infty {n-1choose x-1} int frac{1}{q} frac{1}{(n-1)} q^{n-1} dq\
&=qsum_{n=x}^infty {n-1choose x-1} int frac{1}{q} int q^{n-2} dq dq\
&=qint frac{1}{q} int left[sum_{n=x}^infty {n-1choose x-1} q^{n-2}right] dq dq\
&=qint frac{1}{q} int (1-q)^{-x} q^{x-2} dq dq\
end{align*}

The combinatorics identity I use can be proved using the following argument :
begin{align*}
S(x,q)&=sum_{n=x}^infty {n-1choose x-1} q^n\
&= sum_{n=x}^infty left [ {nchoose x} - {n-1 choose x}right] q^n\
&=sum_{n=x}^infty {nchoose x}q^n - sum_{n=x}^infty{n-1 choose x} q^n\
&= frac{1}{q}sum_{n=x+1}^infty {n-1choose x}q^n - sum_{n=x+1}^infty {n-1 choose x} q^n\
&= left( frac{1}{q} - 1 right) S(x+1, q)
end{align*}

Which means that $S(x,q) = left( frac{q}{1-q} right)^{x-1} S(1,q)=left( frac{q}{1-q} right)^{x-1} sum_{n=1}^infty {n-1choose 0} q^n=left( frac{q}{1-q} right)^x$