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Suppose there is a binomial random variable $X\sim B(n-1,p)$,how to solve the following expectation $$E[(1- b^{X})^{m}]$$ where $b\in (0,1]$ and $m\in \mathbb{N} $ are all constants.
I have tried my best to solve this expectation and got the expression $$\sum_{i=0}^{m}\binom{m}{i}(-1)^{i}(b^{i}\cdot p+1-p)^{n-1}$$.Can I simplify it further?Actually I need an closed-form expression.

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Since $a,b$ are small, could you perhaps round to 3 first terms? Then you need $1 - ma E[b^X] + m(m-1)a^2/2 E[b^{2X}]$... – gt6989b Feb 20 at 5:28
I don't think that's what I want,but thank you anyway. – yyzhang Feb 20 at 6:15
@saz yes, $m \in \mathbb{N}$. – yyzhang Feb 20 at 9:00
One does not solve an expectation, one computes it. – Mariano Suárez-Alvarez♦ Feb 23 at 7:54
@MarianoSuárez-Alvarez I know how to compute this expectation,but i need an expression of it. – yyzhang Feb 23 at 8:20

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