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Let $X \sim binom(n,p)$ be a binomial random variable. I am interested in calculating the expected value of $f(X)= e^{\frac{-a}{bX+c}}$, where $a,b,c$ are positive numbers.

Can someone help me understand how to go about solving this?

I am thinking there might be a way by using probability generating function. But, I can't quite find the PGF of $1/X$ to start with.

EDIT: To give more premise of the problem, I can assume that $n>>1, p<<1$. If still, there is no way to compute the expectation approximately, I am ok with a reasonable upper bound.

wanderer
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  • I suspect this has no closed form solution. You can only expect to obtain some approximation, for example using the CLT for large $n$... – leonbloy Apr 12 '21 at 21:23
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    Slightly related https://math.stackexchange.com/questions/497976/reciprocal-of-a-binomial https://math.stackexchange.com/questions/154060/sum-with-binomial-coefficients-sum-k-1m-frac1km-choose-k – leonbloy Apr 12 '21 at 21:25
  • https://www.wolframalpha.com/input/?i=sum_%7Bn%3D0%7D%5EN+Binomial%5BN%2Cn%5D%28p%5En%281-p%29%5E%7BN-n%7D%29exp%28-a%2F%28bn%2Bc%29%29 wolfram alpha doesn't seem to find any nice closed forms solutions, even using hypergeometric in this case. – Rahul Madhavan Apr 12 '21 at 21:30
  • Maybe if you described the need for finding the expectation, you might get more specific help. For instance, is this for large $n$ and you need something faster than the basic definition: $\sum _{x=0}^n p^x \binom{n}{x} (1-p)^{n-x} \exp \left(-\frac{a}{b x+c}\right)$. Or are you looking for how the expectation changes with increasing values of $b$ given that $a$, $c$, $p$, and $n$ remain constant? – JimB Apr 12 '21 at 22:19
  • @JimB I have added an edit to give more info. – wanderer Apr 13 '21 at 16:24
  • That's good additional information. I suspect that $\exp \left(-\frac{a}{b n p+c}\right)$ will be an upper bound that could probably be made better. A single example is when $a=4$, $b=1$, $c=1/10$, $n=200$, and $p=1/10$, the "almost true" answer is 0.811673 and the upper bound formula gives 0.819546. – JimB Apr 13 '21 at 18:07

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