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Questions tagged [negative-binomial]

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

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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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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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Conditional probability of Negative Binomial R.V. given the SUM of its values

Suppose $\{z_{ij}\}$ are independent Negative Binomial random variables with means $\{\mu_{ij}\}$, with $i=1\dots I$ and $j=1\dots J$. How do you find the (expectation of) conditional probability ...
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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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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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Max of conditional Negative Binomial

Suppose $X|K = w$ is a Negative Binomial with parameters $r$ and $q$. K follows a Binomial Distribution with parameters $m$ and $p$. I want to calculate the expected value of $$Z = max(X_1, X_2,...,...
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Conditional Moment Generating Function of a Negative Binomial

Suppose X conditioned on K follows a Negative Binomial Distribution i.e. $X|K = k_i \sim NB(r-k_i, q)$, where $r$ is a constant and $K \sim Bin(m, q)$. I'm trying to calculate the MGF of X. So far, I ...
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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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Expectation of k-th order statistic of Negative Binomial Distribution

Let $ X_1, X_2,...$ be i.i.d $NB(k,q)$. I am interested in calculating the expectation of their k-th order statistic $X_{k:n}$. From my understanding of order statistics, the CDF of $X_{k:n}$ is given ...
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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}^{\...
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Negative binomial distribution - find the probability that 7 games will be played

Suppose that two teams are playing a series of games, each of which is independently won by team A with probability p and by team B with probability (1-p). The winner of the series is the first team ...
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Evaluate $E(1/X)$ for rv $X$ of the negative binomial distribution.

Let $X$ be a random variable of the negative binomial distribution, of parameters $r\in \mathbb{N},~p\in (0,1)$. Evaluate $E\left(\dfrac{1}{X}\right)$. Attempt. Of course $$E\left(\dfrac{1}{X}\right)...
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Generalised Binomial Series

I am reading an old paper "1971A Generalized Negative Binomial Distribution" by G. C. Jain and P. C. Consul. The paper mainly start from $$(1+z)^n=\sum_{x=0}^{\infty}\frac{n}{n+x\beta}\binom{n+x\beta}...
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Probability you play the piano song at least 8 times before getting it correctly 4 times?

Question: You are working on a difficult passage from a new piece you are learning on the piano.You wish to play it correctly 4 times before calling it a day. If you have a probability of 2/3 of ...
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1answer
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How to show $U$ and $V$ are not independent random variable?

$U$ stands for the number of trials to get the first head, $V$ stands for the number of trials to get two heads. I used hand-waving proof, saying that you could not have the two heads trials without ...
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Conditional expectation value of battery drawing problem

A box contains $a$ batteries of which $d$ are dead. The batteries are tested randomly, one by one. Every time that a good battery is drawn, it is returned to the box. When a dead battery is drawn, it ...
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Distribution of $T_k$ where $T_k - T_{k-1}$ is a Geometric with parameter p

Consider the stochastic process $\{X_n, n=0,1,...\}$ a Bernoulli process with parameter $p$. Let $S_n = \sum_{i=1}^{n}X_i $, with $S_0 = 0$. Define $\{T_k, k=0,1,...\}$ by $$T_k = min_{n}\{S_n >= k\...
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How calculate confidence interval for the mean of negative binomial distribution

How can I calculate a confidence interval for mean negative ofbinomial distribution ? Is there any close formula ? Some help would be appreciated
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Joint Distribution, Geometric Distribution

Let X and Y be IID Geom(p) (independent and identically distributed geometric probability) and $N=X+Y$. Find the joint PMF (probability mass function) of $X,Y,N$. I know the solution to this but I ...
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$\sum_{N=6}^{\infty}\frac{6}{N}\binom{N-1}{5}p^6(1-p)^{N-6}$

Find the sum: $\sum\limits_{N=6}^{\infty}\frac{6}{N}\binom{N-1}{5}p^6(1-p)^{N-6}$ I could not find a way to manipulate $\binom{N-1}{5}$ to get any suitable form here. Note that $\binom{N-1}{5}p^6(1-p)...
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How to prove inequality between probabilities of negative-binomial random variable and geometric variable?

The problem is: Let $X$ be a negative binomial random variable of parameters $r,p$, then $X=Y_1 + Y_2+\ldots+Y_r$, where $Y_j$ ,$j=1,\ldots,r$ are geometric random.variables of parameter $p$. Show ...
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Distribution of a random variable in a coin toss

Amanda tosses a fair coin until she gets $H$. Let $X$ be the number of these tosses. After that she tosses $X$ fair coins, each one until she gets $H$. Let $Y_i$ be the number of tosses for the coin ...
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Sample average from negative binomial probability distribution

My problem is related to sampling from negative binomial probability distributions with unknown true average values, and how confident we can be that the true average of the distribution is above/...
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computing probabilities of an negative binomial random variable

Would appreciate if you could check if I answered the questions correctly: $X$ is a Negative Binomial random variable with the parameters $\frac{1}{2}$ and $r =1,2,3,\ldots$. $Y$ is a random ...
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Probability of binomial n success before m failures?

problem of n success before m failures where binomial probability of success is p has a standard textbook solution as follows $$P = \sum_{k=n}^{m+n-1} \binom{m+n-1}k p^k (1-p)^{m+n-1-k}$$ I am ...
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Confusion about Negative binomial distribution.

I seem to have a bit of confusion about this particular distribution , and I would appreciate if people could help me get past it. My question are as follows. Let $X$ be a discrete random variable. ...
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A forest contains 100 deer. 20 of them have a red tag and…Find the joint pmf of $X$ and $Y$.

Exercise: A forest contains $100$ deer. $20$ of them have a red tag and $80$ of them are untagged. A researcher samples $30$ random deer without replacement. Let $X$ be the number of tagged deer in ...
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Negative binomial expansion problem

So here's the problem:- Show that if x is small, the expression $$\sqrt{(1+x)(1-x)^{-1}}\approx 1+x+0.5x^2$$
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Approximation Binomial theorem “Again 2”

How to prove that if $x$ is nearly equal to 1, then $$px^p−qx^q≈(p−q)x^{p+q},$$ where $p$ and $q$ are any numbers? My try is this: Since $x$ is nearly equal to $1$, put $x=1+h$ or $x=1−h$, where $h$ ...
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Unbiased estimator for negative binomial distribution

Exercise: A biased coin has a probability $p$ that it gives a tail when it is tossed. The random variable $T$ is the number of tosses up to and including the second tail. Show that $\frac{1}{T-1}$ ...
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Why does the negative stay with the fraction after factoring out a -1 when dealing with opposite factors?

So, I understand what I'm supposed to do when coming across opposite factors when simplifying rational expressions. For example: $\dfrac{4-w}{w^2-8w+16}$ simplifies to $\dfrac{4-w}{(w-4)(w-4)}$ So ...
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Geometric to Negative Binomial

I have seen that X ~ G(p) and we have X - 1 ~ NB(1,p) (*) I do not understand the meaning of (*) I definitely know that geometric dist helps to find the number of trials till the first success and ...
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Poisson and Negative Binomial distributions- Mean and Variance total claim size

I am trying to do the following: Let $S$ be the total claim size when the number of claims follow a Negative Binomial Distribution. How can I derive a formula for the $E(S)$ - expectancy and $V(S)$ -...
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What is the probability it will take more than 20 people..?

At airport security there is a bucket with 2 red balls and 8 white balls. For each person, the security guard will select a ball with replacement. People who get a red ball will be ...
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Variance of Negative Binomial Distribution

Most sources say it's $\frac{r(1-p)}{p^2}$ but I can't seem to get it. What am I doing wrong? Let $X$ =$X_1 + X_2 + \cdots + X_r$. $Var(X) = E(X^2) - E(X)^2$ $Var(X_1 + X_2 + \cdots + X_r) = E((...
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Why are there different forms of the negative binomial distribution?

I have found the following two forms of the negative binomial distribution: 1) Let random variable $X$ be the number of failures before $r$ successes are obtained. Then the pmf of $X$ is given by $$...
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Negative Binomial Distribution Question

$10\%$ of new businesses fail within the first year. The records of new businesses are examined until three businesses that failed within the first year are found. Let $X$ be the total number of ...
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How to find the missing terms

Consider the product series: $(x-a)$ $(x^{2} -b)$ .......$(x^{14} -n)$ I want to express the result in this form: $x^{1+2+3+.....14}$ + (some constant)$x^{1+2+3+...14-1}$ +.......so on My ...
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Fitting Count Data with Poisson & NBD

Assume I have a set of count data that I want to fit with a Poisson/NBD If I have data of the form: then it is rather simple...since we can simply use P(X=x). However what would be the approach to ...
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Probability: Negative binomial mode trouble (Exam P SOA 140)

The question states: Each time a hurricane arrives, a new home has a 0.4 probability of experiencing damage. The occurrences of damage in different hurricanes are mutually independent. Calculate the ...
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Non integer successes in negative binomial distribution.

How do we calculate the probability for negative binomial distribution when the number of successes are non-integer? It's easy to calculate when the failures are non-integer by using gamma relation, ...
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Concentration inequality for i.i.d. negative multinomial variables

Let $\mathbf p=(p_0,p_1,\cdots,p_m)$ be a probability vector, such that $p_0+p_1+\cdots+p_m=1$. Let $X_1,X_2,\cdots$ be i.i.d. random variables distributed according to the categorical distribution ...