Questions tagged [negative-binomial]

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

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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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Why does negative binomial not apply [closed]

so there this question that A telephone saleswoman arranges a sequence of interviews of potential customers in order to sell them an insurance policy. She believes that her success rate in completing ...
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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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How to test whether a negative binomial distribution fits my data using the (a, b, 0) methodology?

Question One thousand policies were sampled and the number of claims per policy were recorded in the table below: ...
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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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Mode of Negative distribution

For $NB(r,p)$ where $r$ is the number of successes and the $p$ is the probability of success. Also we have: $$f(x) = {r+x-1\choose r-1}p^rq^{x}$$ I've shown that $f(x)=\frac{(r+x-1)q}{x} f(x-1)$. So: $...
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unbiased estimator for the negative binomial distribution

Let be $P(X=x)= \binom{x-1}{r-1} \theta^r (1-\theta)^{x-r}$ Show that $\frac{r-1}{x-1}$ is an unbiased estimator for 𝜃 My attempt; E$[\frac{r-1}{x-1}]$ $=$ $\sum_{x=0}^{\infty}\frac{r-1}{x-1}\binom{x-...
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Variance of negative binomial distribution - proof

I am testing the variance of a negative binomial distribution, but I have problems in the derivative. $$ V(x) = E(x(x-1))+E(x)-E(x)^{2} $$ $$ E(x(x-1)) = \displaystyle\sum^{\infty}_{\substack{x=0}} x ...
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Two Series Encountered in the Canonical Representation of Negative Binomial Distribution

When I calculate the Lévy-Khintchine canonical representation of the negative binomial distribution, I need to find the sum of the following series $$ \gamma_n=n\sum_{k=0}^{\infty}\frac{k}{1+k^2}\left(...
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binomial identity seemingly illogical and impossible. Is there any way it could be true?

There is binomial expression(s) written as $$\sum_{n\geqslant0}\frac{(-3n+2k-3)n!^2}{2(2n+1)(k-1)!^2(n-k+1)!^2 \binom{2n}{n}}=\begin{cases} 0 & \text{if $k=0$,} \\ -1 & \text{if $k\geqslant1$,...
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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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Confusion about relationship between geometric and negative binomial RV's

(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, ...
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What Variance equation is this textbook using?

I'm doing a question on a Gamma/Poisson mix, which turns into a negative binomial. What I don't understand is the answer to the question - most of the answer makes sense to me (normal approximation ...
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Negative Binomial Distribution: impossible probabilities?

From what I understand, negative binomial is $P(X=k$ number of trials until $r^{th}$ success) From this chart, For the red graph $r = 10$, $P(X=k$ number of trials until $10^{th}$ success) For all $X&...
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Mean of the sum of two independent geometric distributions with different probabilities of success

Let $X \sim Geom(p_1), Y \sim Geom(p_2)$ be two independent geometrically distributed random variables with probabilities of success $p_1, p_2$ respectively. I want to find the mean of the sum of ...
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mean and variance formula for negative binomial distribution

The equation below indicates expected value of negative binomial distribution. I need a derivation for this formula. I have searched a lot but can't find any solution. Thanks for helping :) $$ E(X)=\...
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What is the expected number of trials before x successes with variable probability?

I am trying to calculate the probability of reaching n successes on a trial, knowing that a failure will increase success chance. In simpler term, what is the mean number of trials before we reach $x$ ...
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Expected number of Bernoulli trials before at least s successes and f failures

Let a random variable $X$ be the number of independant Bernoulli trials needed to reach $s$ successes and $f$ failures when the probability of success is $p$. We therefore stop trials when we have $s$ ...
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Probability question that links between $NB$ and $Gamma$ distributions with limits.

Let $\lambda>0$, and $ m\ge 1$, $m\in\mathbb{N}$. For every $n\ge 1:$ $Y\sim NB(m, \frac{\lambda}{n})$, we define: $\hat Y_n=\frac{1}{n}Y_n$. Show that for every $t\ge 0$: $F_{\hat Y_n}(t)\to F_{\...
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Non-probabilistic proof for this infinite series identity related to the generalized negative binomial distribution?

Using the alternative "Poisson-like" parameterization of the generalized negative binomial distribution in terms of its mean $\lambda$, it follows that the PMF can be written $$ \frac{\Gamma(...
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Sum of random number of random variables is negativ binomial distributed.

I have problems to solve the following: if $N$ is Poisson distributed with $\lambda=-r\ln(p)$ und $Y=\sum_{k=1}^N X_k$ is negativ binomial distributed with parameters $r$ und $p$, what are $X_k$? We ...
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2 votes
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How to derive Negative Binomial process from Poisson process?

I am trying to understand NB process, and how it can be derived from Poisson process. Zhou & Carin states that: "By placing a gamma prior with shape $r$ and scale $\frac{p}{1−p}$ on $λ$ as $m ...
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Binomial expansion of negative exponent in descending powers of x [closed]

Expand ${(1+x)}^{-2}$ in descending powers of x including the term ${x}^{-4}$
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How to split a pot of $100 when a game of flipping coins until first person gets 10 heads is interrupted.(A variation of the points problem)

Question : Andy and Beth are playing a game worth $100. They take turns flipping a penny. The first person to get 10 heads will win. But they just realized that they have to be in math class right ...
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Question regarding Probability of Dice [closed]

A pair of dice is thrown 180 times in a row. Find the probability that the event is 25 or more times I should define random variable $X$ but is this binomial distribution or negative binomial ...
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Confusion on how extended binomial theorem works

So I just finished learning the standard binomial theorem and I've just come across the extended (newtons binomial theorem). As expected I am completely baffled about how it works I do not understand ...
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1 answer
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A negative binomial distribution for two random variables?

In a sequence of Bernoulli trials with success chance $p$, the probability that we will see $k$ failures before the $r$-th success is given by the negative binomial distribution (NBD) with pmf: $$ Pr(...
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How to taylor expand $1/(1- \frac{k}{r})$ where k is some constant

Hi i am doing some integrations and there came across one step that i couldn't understand. The step is as follows: $$\frac{dr}{1 - \displaystyle\frac{k}{r}}=dr\left(1 + \frac{k}{r-k}\right)$$. I don't ...
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An electronic scale in an automated filling operation stops the manufacturing line after five overweight packages are detected.

An electronic scale in an automated filling operation stops the manufacturing line after five overweight packages are detected. Suppose that the probability of an overweight package is 0.05 and each ...
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How was this simplification done?

\begin{align*} \begin{pmatrix} -2\\ n\end{pmatrix}&=\frac{(-2)(-3)(-4)\dots(-2-n+1)}{n!}\\[5pt] &= (-1)^n\frac{2\times3\times4\times\dots n(n+1)}{n!}\\[5pt] &=(-1)^n(n+1)\\[...
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How to evaluate $ \sum_{n=r}^{\infty} n^2 \binom{n-1}{r-1} p^r (1-p)^{n-r}$?

I want to evaluate the following sum $$ \sum_{n=r}^{\infty} n^2 \binom{n-1}{r-1} p^r (1-p)^{n-r} $$ where $r \in \mathbb{N}$ and $0<p<1$. Using WA I know that the sum evaluates to $\frac{r(r-...
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How to compare binomial distributions with different number of trial?

I want to know which event has the best score (or luck or odd) $p$ (prob of success) is equal with all event. There are givne set of events, that is (number of try, number of success) $$\mathrm{event1}...
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Negative binomial (and multinomial) extension to the minimum(failures,successes)

The negative binomial distribution defines the probability of N draws to obtain K successes. The negative multinomial gives the joint over k1...km given k0 successes. I need to find the PMF over n ...
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Lower index of binomial coefficient becoming negative upon solving

The question While solving this question by using the property: r nCr = n n-1Cr-1 The value of lower index becomes negative which is invalid How do I proceed?
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Marginal Dirichlet Negative Binomial Distribution and the Multinomial Inverse Polya Urn

I have the following 'URN-like' problem - assume an urn the contains balls with m different colors. As in the standard Polya scheme, every time a ball is sampled, it is returned the urn in addition ...
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Negative Binomial Series

On this webpage, the expansion of the negative binomial series is given below. $$ (x+a)^{-n} = \sum^\infty_{k=0} (-1)^k \begin{pmatrix}n + k - 1\\k\end{pmatrix} x^ka^{-n-k} $$ when $|x| < a$. My ...
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Hypothesis testing with the weighted average of negative binomial distributions

I am trying to do a statistical hypothesis testing between two random variables $Y^{(1)}$ and $Y^{(2)}$: $Y^{(1)} = \frac{\sum_{i=1}^{n}w_i^{(1)}X_i^{(1)}}{\sum_{i=1}^{n}X_i^{(1)}}$ $Y^{(2)} = \frac{\...
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Uniformly minimum variance unbiased estimator for negative binomial distribution

I'm working through an old qualifying exam and I feel like I'm so close to the answer. Given a random sample of size n from a negative binomial distribution with parameters (r,p) I need to find a UMVU ...
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-2 votes
1 answer
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Minimum Variance Estimators that Attain the Cramér-Rao lower bound

The probability mass function of a negative binomial random variable $X$ is given by $$P(X=x\mid\theta)= {r+x-1 \choose x}(1-\theta)^x\theta^r$$ and $\Bbb E(X\mid \theta)=\frac{r(1-\theta)}{\theta}$. ...
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1 answer
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E[1/(Y+1)] of negative binomial distribution [closed]

There was an exercise in class asking to solve for the expectation E[1/(Y+1)] when Y is a negative binomial random variable (number of failures). How do I solve this? Whatever transformation or ...
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is $\frac{Γ(r+x)}{Γ(r)Γ(x+1) }$ the same as ${r+x-1 \choose x}$?

is $\frac{Γ(r+x)}{Γ(r)Γ(x+1) }$ the same as ${r+x-1 \choose x}$ ? I have two pmfs $$P(X=x\mid\theta)={r+x-1 \choose x}(1-\theta)^x\theta^r $$ and $$P(X=x\mid\theta)=\frac{Γ(r+x)}{Γ(r)Γ(x+1) }(1-\theta)...
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Quick question about the definition of a random variable in a baseball pitches example

I'm just wondering, in the last sentence of the third paragraph, why can X not be larger than 6? Is that because the number of innings is 9? I'm really confused, I thought X goes from 3,4,5,6,...to ...
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Expectation of negative binomial distribution

Given $X \sim \text{NBin}(n,p)$, I've seen two different calculations for $\mathbb{E} (X)$: \begin{align*} &1. \mathbb{E} (X) = \frac{n}{p}, \quad \text{or}\\ &2. \mathbb{E} (Y) = \frac{n(1-p)}...
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What is the size of a multinomial?

The answer to this question uses the phrase "multinomial of size". What is the definition of the size of a multinomial? They are using a negative multinomial.
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PGF of negative multinomial expansion

I have found the formula for the Probability Generating Function of negative multinomial distribution in Definition 8.1 of this chapter (https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118445112....
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3 answers
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When rolling a die, what is the probability that a 4 has appeared at least 3 times by the 15th roll?

Now I've managed to figure out the probability that the 3rd time a 4 appears is on the 15th roll using a negative binomial distribution. But the at least is totally screwing with me in this variation ...
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1 vote
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Negative multinomial expansion

I would like to know the standard form of the negative multinomial expansion i.e. $(x_1 + x_2 + \ldots + x_p)^{-n}$. I understand that I can probably derive something by applying the negative binomial ...
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