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Questions tagged [negative-binomial]

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

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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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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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Claim size-Conditional probability-Negative Binomial distribution

Hi can anyone help me on this? The number of claims for a portfolio for a one week period has a Negative Binomial distribution with $p=0.4$ and $k=6$. How can I calculate the following: Given that ...
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How to derive a modified formula for Total Claim size Negative Binomial distribution

can anyone help me on this? How can I derive a modified formula for E(S) and Var(S), where S denotes the total claim size, when the number of claims per year has a Negative Binomial distribution with ...
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Probability that it take atleast “i” flips to get the third head appears?

Problem: Flip a fair coin until the third head appears, and then stop right after that flip. What is the probability that it took you atleast "i" flips to accomplish this? My first apprach: Taking ...
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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 ...
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What is my error in calculating expected value of a negative binomial random variable?

We perform Bernoulli trials until $r$ failures ($P.(\text{fail}) = u$) occur. What is the probability that $k$ successes occur? \begin{align*} p(k) = {r + k - 1 \choose k} u^r (1 - u)^k \end{...
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Problem with Negative Binomial Distribuition

Suppose two players A and B are trying to hit a basket of basketball. The probability that A hit at a given pitch is p, and he insists on the throws until he hits r times. The probability that B ...
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Sum of two random variables ( negative binomial distribution )

Let $X,Y$ be two independent negative binomial distributed random variables. $X$ ~ $NB(r,p)$ and $Y$ ~ $NB(s,p)$ Show that: $ X+Y $ ~ $ NB(r+s,p) $. Remark: So where I'm stucked? I failed to show ...
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Tricky question on binomial [duplicate]

Let's say there's a series of the form $$S=\frac{1}{10^2}+\frac{1\cdot3}{1\cdot2\cdot10^4}+\frac{1\cdot3\cdot5}{1\cdot2\cdot3\cdot10^6}+...$$ Now i had written the rth term as $$T_r=\frac{1\cdot3\...
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A couple want to have one girl and one boy. What is the expected number of children they will have? [closed]

A couple want to have one girl and one boy. What is the expected number of children they will have? I know that the answer is 3 and that the question relates to the Negative Binomial distribution but ...
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Negative binomial distribution interpreted as a waiting time

I'm reading Achim Klenke's text on probability theory, and a few different distributions (with applications) have been introduced. I'm confused by the explanation given of the negative binomial ...
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Negative binomial distribution example

I apologize if this is too simple a question. One of the exercises in my probability textbook says "A dice is rolled until the first time T that a 6 appears. Find P(T > 3)" The answer given is (5/6)...
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Setting the range on Negative Binomial and Geometric distributions

Unlike other discrete distributions I find that setting the range and solving for summation in negative binomial distributions to be quite tricky. Here's what I mean via a sample problem: some ...
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Negative Binomial with $4$ white faces before $3$ black faces

Suppose that a fair $6$-sided die having $2$ black faces and $4$ white faces will be rolled repeatedly. What is the probability that $4$ rolls resulting in a white face occur before $3$ rolls ...
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Binomial theorem and series

I want to confirm some points about binomial series. The expression must be of the form$(1+x)^n$. $x<1$ The part where I am confused is that while finding derivatives through first principle , i'...
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Covariance of Geometric Distribution and Negative Binomial Distribution

Let independents $X_1,...X_n \thicksim Bernoulli(1,p)$ and $W_r$ refer to total number of trial until the r-th success, which means $W_r \thicksim Negbin(r,p)$ then I need to evaluate $Cov(W_1,W_r)$. ...
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Negative binomial as limit of the negative hypergeometric

It is known that the hypergeometric distribution can be approximated by the binomial distribution (it is also shown here Proof that the hypergeometric distribution with large $N$ approaches the ...
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Distribution of sum of correlated negative binomials

Is sum of two negative binomial variables also distributed as negative binomial if they are non-idependent and correlation is $ < 1$? Or not?
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Probability that 4 boxes are purchased?

The probability that a randomly selected box of a certain type of cereal has a particular prize is 0.2. Suppose you purchase box after box until you obtained two of these prizes. What is the ...