# Questions tagged [negative-binomial]

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

139 questions
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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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### 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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### 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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### 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 ...
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 ...