# Questions tagged [negative-binomial]

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

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### Proving that $T_{n}(X)=\sum_{i=1}^{n}X_i$ is a sufficient statistic for $p$ given a sample of i.i.d random variables

I am asked to show that the statistic $T(X):=\sum_{i=1}^{n}X_i$ is a sufficient statistic for $p$, where $X_i\sim Geom(p)$ are i.i.d random variables. Given a sample $x=(x_1,x_2,\dots,x_n)$ I have to ...
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### I am trying to understand what an expectation is

Let $\{B_j(p) | j \geq 1\}$ be a sequence of Bernoulli trials or an i.i.d. sequence of Bernoulli$(p)$ random variables, where the win probability is $$p = P\{B_j(p) = 1\} = E[B_j(p)].$$ Now define $X$ ...
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### Is binomial distribution measuring combinations or successes? [closed]

So I think I understand what binomial coefficients are, "K number of combinations that can be made from set N". However, I keep seeing examples demonstrating K being the number of successes ...
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### Question on comparing two probability spaces

Let's assume we have two discrete probability spaces $(\Omega_1,P_1)$ which describes experiment $1$ and $(\Omega_2,P_2)$ which describes experiment $2$. We know the probability $P_1$ but we don't ...
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### Question on solution of matchbox problem

Suppose a mathematician carries two matchboxes at all times: one in his left pocket and one in his right. Each time he needs a match, he is equally likely to take it from either pocket. Suppose he ...
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### Question on applying negative binomial distribution

Let's assume we conduct an experiment which generates two outcomes: success $A$ or failure $B$. The experiment never generates more than $N$ failures. Both $A$ and $B$ have the same probability. After ...
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### Probability space of Banach matchbox problem

Suppose a mathematician carries two matchboxes at all times: one in his left pocket and one in his right. Each time he needs a match, he is equally likely to take it from either pocket. Suppose he ...
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### Is there a formulaic relation between the accuracy of the approximation and the number of terms used in the binomial expansion

So I am aware that with the increase in number of terms the approximation becomes more accurate but I wished to know if I have a binomial of the form say, $$f(x) = (1-Rx)^{-n}$$ where $R$ is a real ...
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### When is the binomial expansion with negative exponent valid for polynomials with complex numbers

So for a normal binomial expansion with negative exponent, say $$f(x) = (1 - 2x)^{-1},$$ we know it is valid for $$-1/2 < x < 1/2.$$ But what is the case for complex numbers? I was able to see ...
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### How long it takes to roll 6 6's in a row

Someone is sick and extremely bored. To pass the time, he decides to roll a $6$-sided die until he gets $6$ $6$'s in a row. If it takes him $5$ seconds for every roll of the die, how long is he ...
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### Minimum Variance Unbiased Estimator for a Negative Binomial distribution

Given a negative binomial distribution parameterized by an unknown mean $\mu$ and known dispersion parameter $\alpha$ (so that the total variance is of the form $\mu + \alpha*\mu^2$), what is the ...
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### Generalizing the sum for the binomial expansion of a number with negative exponent

It is known that $$\sum_{n=0}^{\infty} \binom{z+n-1}{n}(-1)^{n}x^{n} = (1+x)^{-z},$$ but what about sums of the form $$\sum_{n=t}^{\infty} \binom{z+n-1}{n}(-1)^{n}x^{n}?$$ Is there an explicit ...
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### Negative Binomial mean

Let $X\sim BN(r,p)$. If I try to compute E[X] through the Moment Generating Function I get the following: \begin{aligned} E[X] &=\left.\frac{d}{d t} M_{X}(t)\right|_{t=0} \ &=\left.\frac{d}{d ...
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### Understanding binomial theorem with negative integer of n [duplicate]

I’m a student studying mathematics and I understood how the binomial theorem with positive integer of n works. But I couldn’t really understand the progress of how the binomial theorem with negative ...
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### Tim plans to attempt exam until pass. Chance of passing each time is $20\%$. Find probability that he needs to attempt atleast $2$ & atmost $5$ times

Question : Tim is in no hurry to graduate. He intends to write his last exam until he passes it, but he is too lazy to study. He estimates that the chance of passing the exam each time is $20\%$. ...
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### Probability mass function of alternative Negative binomial distribution

I have a random variable $X$ defined by the sum of $k$ independent geometric distributions ($Y$) with parameter $p$, which makes it a negative binomial random variable $NB(k,p)$. The probability mass ...
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### Why does Negative binomial expansion have infinite terms

Why does negative binomial expansion have infinite number of terms and not equal to the example given below Why is $(x+a)^{-2}$ not equal to $\frac{1}{x^2 + a^2 + 2ax}$? ?
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### How to make negative binomial distribution model based on data?

I have a set of data that has actual frequency. I have difficulty to find the method of building the negative binomial distribution model. How to relate the parameters in negative binomial ...
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### Why use Negative Binomial distribution to model count data?

According to Wikipedia: https://en.wikipedia.org/wiki/Negative_binomial_distribution In probability theory and statistics, the negative binomial distribution is a discrete probability distribution ...
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### Is it possible to simplify this binomial expression?

There is binomial expression(s) written as $$\sum_{n>0}\frac{(-3n+2k-3)n!^2}{2(2n+1)(k-1)!^2(n-k+1)!^2 \binom{2n}{n}}=0\; if\; k=0\; or= -1\; if\; k>0$$ which simplifies to \sum_{n>0}\frac{(...
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### Expected value of a continous generalization of the negative binomial distribution

Let the random variable $X$ be the number of independent Bernoulli trials needed to reach $r$ failures when the probability of success is p. Then $X$ follows a negative binomial distribution \begin{...
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(Geo for geometric random variable, NB for negative binomial) Let $X_1,\ldots,X_n\sim\text{Geo}(1/n)$ be i.i.d and let $X=\sum_{i=1}^nX_i$, I saw on wikipedia that $X\sim\text{NB}(n,1/n)$. However, ...