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Suppose a random variable $X$ is distributed as $\operatorname{NB}(\mu, \theta)$, and its mass is as follows $$ \mathrm{P}(X = y) = \binom{y + \theta - 1}{y} \left(\frac{\mu}{\mu + \theta}\right)^{y}\left(\frac{\theta}{\mu + \theta}\right)^{\theta}.$$ Does anyone know how to calculate the expectation of $1 / X$ in this kind of definition of negative binomial distribution?

If there is no explicit expression for $\mathrm{E} (1 / X)$, I wonder whether exists some lower bounds for $$\mathrm{E} \left[ \frac{X}{(X + c)^n}\right],$$ where $c$ is some positive constant and $n \in \mathbb{N}$.

Thanks so much!

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  • $\begingroup$ I believe this question would be more appropriate in math.stackexchange.com. Moreover, I think the title does not represent well your final question. $\endgroup$
    – Kernel
    Commented Nov 2, 2020 at 9:43
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    $\begingroup$ Isn't $X = 0$ with positive probability? $\endgroup$
    – Mateusz Kwaśnicki
    Commented Nov 2, 2020 at 9:51
  • $\begingroup$ @Kernel Thanks. My final question is finding a lower bound for $\mathrm{E} \left[ X/ (X + c)^n\right]$. The reason for my first part in the question is that: by Jensen inequality, $\mathrm{E} \left[ X / (X + c)^2\right] \geq \left[ \mathrm{E} X + 2c + c^2 \mathrm{E} (1/ X)\right]^{-1}$, then finding a upper bound for $\mathrm{E} (1/X)$ is enough. $\endgroup$
    – 香结丁
    Commented Nov 3, 2020 at 5:54
  • $\begingroup$ @Kernel Oh! You said the title. Got it! Thanks so much~ $\endgroup$
    – 香结丁
    Commented Nov 3, 2020 at 5:56

1 Answer 1

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Mathematica answers your second question for concrete values of $n$ (e.g. $n=3$) by

Mean[TransformedDistribution[X/(c + X)^3, X \[Distributed] NegativeBinomialDistribution[\[Mu], \[Theta]],Assumptions->c>0]]

$$-\frac{(\theta -1) \mu \theta ^{\mu } \, _4F_3(c+1,c+1,c+1,\mu +1;c+2,c+2,c+2;1-\theta )}{(c+1)^3} $$

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  • $\begingroup$ Thanks. But it seems that it is also involved to find a lower bound of the expectation from this formula. $\endgroup$
    – 香结丁
    Commented Nov 3, 2020 at 6:01

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