# Questions tagged [negative-binomial]

Questions about the negative binomial distribution, a discrete probability distribution.

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### Calculating $E\left[\frac{r}{X}\right]$ where $X$ has a negative binomial distribution

Performing a series of Bernoulli trials until $r$ (positive known integer) successes. Denote $p$ as the probability of success, $q = 1-p$ as the probability of failure and $X$ a random variable counts ...
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### Confusion on the proof and meaning of negative binomial random variables

Going through the book Introduction to Probability, Statistics and Random Processes, I stumbled across an interpretation of negative binomial random variables and its proof. Suppose I have a coin ...
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### the generated function of pulling a ball from a bin of balls with size that it is disturbuted by NP(2,p)

let's say that I got a bin of balls, which its size is distributed negative binomially: NP(2,p). And let's say that each of the balls is numbered from 1 to x-1. What is the generated function of Y: ...
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### Binomial expansion of negative exponent

Stuck on this binomial result doing gravitation chapter in physics the expression is $$(1+x)^{-2}=1-2x$$ provided $x$ is so smaller than $1$ . My questions is Why and second If I want to ...
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### Showing $E(X^k)=\frac{r}{p}E(Y-1)^{k-1}$ where $X,Y$ are negative binomial

Let $X\sim\text{NB}(r, p)$. (a) Show that $E(X^k)=\frac{r}{p}E(Y-1)^{k-1}$ where $Y\sim\text{NB}(r+1, p)$. (b) Use this result to find $E(X^2)$. I am having trouble specifically with part ...
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### MATLAB/EXCEL Inverse Cumulative Distribution Function (ICDF) for Negative Binomial Distribution

I have some old MATLAB code that uses the ICDF() function call (inverse cumulative distribution function). I need to translate the following into an equivalent ...
### Explanation for binomial sums $\sum_{n=0}^{\infty} \binom{-4}{n-1} (-1)^{n-1} x^n = \sum_{n=0}^{\infty} \binom{-4}{n} (-1) x^{n+1}$
I was looking at some negative binomial coefficient problems and I stumbled upon this explanation \sum_{n=0}^{\infty} \binom{n+2}{3} x^n = \sum_{n=0}^{\infty} \binom{n+2}{n-1} x^n= \sum_{n=0}^{\...