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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23 views

Relation Between Binomial and Negative Binomial

I have come across a question in a text book that I can't seem to figure out. Firstly, Let X be the number of faulty items in a batch of 10. What is the distribution of $X$, and what is $P(X = k)...
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29 views

How to scale a negative binomial distribution?

I posted a variation of this question on Cross-validated, but did not get any answer, so I hope someone can help me over here. A bit of background first. I have implemented a neural network for time ...
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26 views

Testing if coefficients are statistically significantly different across models

I will be building two zero-inflated negative binomial (ZINB) regression models, where each model is aiming to predict different disease count outcomes based on the exact same independent variables ...
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1answer
45 views

Expected number of rounds for algorithm to terminate

An algorithm terminates when the input size is less than $1$. For each iteration, there's $1/2$ chance that the input size gets halved, and $1/2$ chance it stays the same. What's the expected number ...
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How do you code Negative Binomial Distribution in R

How do I code the following questions in R? When a machine is improperly adjusted, it has probability 0.105 of producing a defective item. Each day, the machine is run until 3 defective items are ...
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21 views

How to show Lagrange's remainder of the negative binomial expansion tends to zero?

Giving a binomial $(1-x)^{-n}$, it is well known we can expand it like like: $$1 + \binom{n}{1}x + \binom{n+1}{2}x^2 +\ ...$$ I am trying to understand why this expansion converges with $(1-x)^{-n}$ ...
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23 views

Binomial distribution considering starting-a-set-advantage

I am making a model in Matlab that calculates the winning probability of darts player A against darts player B with help of a paper (see bottom line). I have been able to implement the calculation of ...
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36 views

Negative Binomial Problem

I found this question while going through a book: In a casino game Ruben rolls a die and whenever a one or a six is rolled he receives a token. The game ends when Ruben has received y tokens; ...
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36 views

What is the probability of event after $n$ day

I have the following question: In the monsoon season, what is the probability that we will have a sunny day after a series of rainy days, or the opposite what is the probability of rain in summer ...
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43 views

How to prove that $\sum_{k=0}^{\infty}\binom{k+r}{r}p^r(1-p)^k=\frac 1p$?

I want to prove $$\sum_{k=0}^{\infty}\binom{k+r}{r}p^r(1-p)^k=\frac 1p$$ for $p\in (0,1)$ and $q=1-p$, Mathematica told me it's an identity, but I have no idea to deal with the binomial coefficient, ...
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60 views

MLE of negative binomial

The question is: Given a negative binomial distribution of $f(x;\theta) = \frac{4(x+1)}{(2+\theta)^{x+2}}\times\theta^x$, find the MLE for n independent random variables $X_{1}, ..., X_{n}$, and ...
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Marginal probability mass function of bivariate negative binomial

define $$P(X=x,Y=y) = {(x+y+k-1)!\over x!y!(k-1)!}p_1^xp_2^y(1-p_1-p_2)^k$$ the bivariate negative binomial distribution. I am interested in the marginal probability mass function of X. After ...
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Negative Binomial Distribution - Understanding formula inputs

I'm trying to understand the solution to the question below and it does not seem intuitive. So trying to understand how to derive the inputs Q: A single fair-dice is rolled repeatedly. a) What is the ...
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52 views

Negative Binomial Conditional Expectation [closed]

If I have a probability $.2$ that any given customer will order a hamburger at a restaurant, and I know the last $2$ customers ordered a hamburger. What is the expectation for the the number of ...
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15 views

Discrete Variable Distribution: Binomial vs Negative Binomial Distribution

I am self learning the discrete random variable distributions, and I think I have some problems understanding the difference between these two distributions. For what I understand at this stage, I ...
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8 views

Random Variable Generator of a Negative Binomial Distribution

Develop a method to generate a negative binomial distribution with parameters p and k... Im really confused on this one, I have used the convolution method to get the Normal and Erlang random ...
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Finding posterior for Negative binomial likelihood and Beta prior

If X = (X1, . . . , Xn) is a sample from Negative Binomial(m, θ) and θ ∼ Beta(α, β), I get the posterior for θ as beta Be(α + mn, β + xi). Suppose, The number of failures until the 4th success in a ...
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1answer
29 views

Why is this adjustment of the negative binomial distribution true?

SECOND-YEAR PROBABILITY: For iid trials with events $F,S$ and $P(S)=p$, what is the probability $y$ trials will occur before the $r$th success? This is a question from Wackerly, Mathematical Stats ...
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25 views

Maximum Likelihood Estimation for a Sum of Two Negative Binomials

I'm trying to get the MLEs of the parameters of a sum of two negative binomials. There are 5 parameters to estimate and I have managed to get the MLEs of 3 of them. The part of the log-likelihood ...
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23 views

random natural number from range using negative binomial

How can you extract a random number from a range using the negative binomial distribution? For example: the range of possible result values is [1,10] the probability of success is 0.2 I really don'...
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54 views

Expectation of Negative Multinomial Distribution

If a trial consists of throwing an n-sided fair die having numbers a1,a2,a3...,an on its faces. What will be the expected number of trials required before we get atleast k1 times a1, k2 times a2,.......
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50 views

Sum of Two Pascal Random Variables

I'm having trouble finding the PMF of a sum of pascal random variables. The problem stated is: Let $X \sim Pascal(m,p)$ and $Y \sim Pascal(l,p)$ where X and Y are independent of each other. Let $Z = ...
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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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41 views

Distribution of Poisson random variable with scale parameter distributed as Negative Binomial

I would like a closed form solution to $$ P(n):=\sum_{m=0}^\infty \textrm{NegBin}_{r,p}(m)\textrm{Poisson}_{m\lambda}(n)\\ = \sum_{m=0}^\infty \left(\begin{array}{c}{m+r-1} \\ {m}\end{array}\right) \...
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34 views

Calculating mean and veariance of Negative Binomial variable using the parameterization of $NB(r, \mu)$

I am trying to calculate the mean and variance of the negative binomial variable using the parameterization $NB(r, \mu)$, where $\mu$ is the mean of the variable. That is, I am trying to proof that $$...
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155 views

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 ...
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Identifying, Describing a Binomial-type Distribution

Fix $p \in (0, 1), N, M \in \mathbf{N}_{\geqslant 1}$. It is well-known that if $X \sim \textbf{Bin} (N, p)$ and $Z \sim \textbf{Bin} (M, p)$, then $Y = X + Z$ is marginally distributed as $Y \sim \...
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Connecting two interpretations of the negative binomial distribution

In my probability course, my professor derived the negative binomial distribution by reasoning about the probability that the time of the $k$-th success, $T_k$, takes some value $n$. If $p$ is ...
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Zero - modified negative binomial in R

So, I am trying to fit a zero - modified negative binomial distribution to my dataset in R. I already fit negative binomial distribution using the following code: ...
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1answer
63 views

Probability and Statistical Modelling Proof (Negative Binomial into Poisson)

The Negative Binomial RV $X$ models the number of trials until the $r$-th success in a sequence of independent Bernoulli Trials with probability of success $p$ in each trial. So, if $q = 1 - p$, $$P(...
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Need help expanding this negative binomial expanison

I am an A level Student studying for my papers and this is a question from a past paper...the original equation is $$\frac{1}{\sqrt{1+x}+\sqrt{1-x}}$$ which part a says to convert to the following ...
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1answer
108 views

Domain limitations on generating function for Legendre polynomials

The generating function for legendre polynomials is $$\frac{1}{\sqrt(1-2ut+u^2)}=\sum_{i=0}^\infty u^iP_i(t)$$ where $P_i$ is $i^{th}$ legendre polynomial however for binomial expansion of left hand ...
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Does this random vector follow a multinomial distribution?

Suppose a sample $X_1, X_2, ...$ follows a Binomial distribution with parameters $m \in \mathbb N$ and $\pi \in (0, 1)$. Denote \begin{equation} Z_k := \sum_i I(X_i = k) \quad \text{for} \quad k = ...
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2answers
235 views

probability of penalty (in soccer) being scored or being missed

I have the following assumptions. I am expecting 0.4 penalties in a match on average, and I am assuming that penalties follow a Poisson distribution. The probability of a penalty being converted is 82%...
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What is the expectation and variance of a negative binomial distribution NB(r,p)?

This information can easily be obtained, but the notation that I am using is different. I am looking at the number of trials (k) it takes to obtain r successes, given a certain probability p. This is ...
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Finding the expected value of a negative binomial distribution with two success indications

Flip a coin until a head appears or until the fourth trial. Let $X$ be the number of coin tosses. What is $E(X)$? ...
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How is the random variable $X=\max\{x_1,\dots,x_n\}$ related to the negative binomial distribution

Consider a jar with $N$ indexed balls from $1$ to $N$ and the variable $X=\max\{x_1,\dots,x_n\} $ where $n<N$ is the number of balls we took out without replacement. I found that $P(X=k)=\begin{...
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A Summation from A Generalized Negative Binomial Distribution

I am reading Jain and Consul "1971A Generalized Negative Binomial Distribution". The key identity of this generalised negative binomial distribution is (slightly different version): $$(1-\alpha )^{-n}...
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Find the range of p such that team A has the advantage in a best four of seven series.

Two teams, A and B, are playing a series of games. Assume probability that A won a game is p result of a game will not affect result of the next game Find the range of p such that team A has the ...
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Calculating Expected number of stages

An urn has m black balls. At each stage, a black ball is removed and a new ball that is black with probability $p$ and white with probability $1-p$ is put in its place. Find the expected number of ...
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Which distribution do I need to use?

In a shop, the average customers per 5 minutes is 3. What is the probability that the shopkeeper has to wait at least 6 minutes before the second customer walks in. I don't know which distribution I ...
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Binomial distribution: what is the probability of getting exactly 3 women from a draw of Y = 1 to 10?

So my question is We choose a certain number Y of different people what is the probability of getting exactly 3 women from a draw of Y = 1 to 10 ? what is the probability of getting at least more ...
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Computation of the expectation $E(2^{X-2})$ for $X$ negative binomial

If I have a coin with $\ 0.6$ probability of getting $\ H $and I throw it until I get $\ H $ for the second time. If $\ Y $ is the number of $\ T$ I get and $\ 2^Y $ is my revenue. how do I ...
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31 views

Let n belongs to +ve integer and $(1+x+x^2)^n=\sum_{r=0}^{2n} {a_rx^r}$ prove that: $a_r=a_{0<r<2n}$

Let n belongs to +ve integer and $$(1+x+x^2)^n=\sum_{r=0}^{2n} {a_rx^r}$$ prove that: $$a_r=a_{2n-1},{0<r<2n}$$ as well as prove that $$\sum_{r=0}^{ n-1} a_r=\frac{1}{2}(3^n-a_n)$$. I tried to ...
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CDF of a random variable with mulitple conditions

I am trying to solve a particular problem involving multiple conditions on a random variable. We have $X_1, X_2....$ such that $X|K = k_i$ is a negative binomial with number of success given by $max(...
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56 views

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}^{\...