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Questions tagged [negative-binomial]

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

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Proving that $T_{n}(X)=\sum_{i=1}^{n}X_i$ is a sufficient statistic for $p$ given a sample of i.i.d random variables

I am asked to show that the statistic $T(X):=\sum_{i=1}^{n}X_i$ is a sufficient statistic for $p$, where $X_i\sim Geom(p)$ are i.i.d random variables. Given a sample $x=(x_1,x_2,\dots,x_n)$ I have to ...
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I am trying to understand what an expectation is

Let $\{B_j(p) | j \geq 1\}$ be a sequence of Bernoulli trials or an i.i.d. sequence of Bernoulli$(p)$ random variables, where the win probability is $$p = P\{B_j(p) = 1\} = E[B_j(p)].$$ Now define $X$ ...
Statnerd's user avatar
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162 views

Binomial identity?

$${n+k-1 \choose k}=\sum_{m=1}^{min(k,n)}{k-1 \choose m-1}{n \choose m} $$ Is there a simple way to demonstrate this equality? Context These are two ways of expressing the $x^k$ coefficients in $(1+x+...
Older Amateur's user avatar
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Choosing k positions among n with no more distance d each other

Let $\binom{n}{k}_{<d}$ the number of combinations of k elements among [1,n] with constrained spacing : no element can be at distance d or more from its successor. $$\binom{n}{k}_{<d} = \sum_j (-...
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Combinatorial Identities in Negative Binomial mgf derivation [closed]

I'm trying to understand how to derive negative binomial mgf and came accross this discussion Deriving Moment Generating Function of the Negative Binomial? I can't understand the step that assumes, $$\...
HIken's user avatar
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Negative Binomial Distribution questions

In 2016, Red Rose tea randomly began placing one of ten English porcelain miniature figurines in a l00 bag box of the tea, selecting from ten figurines in the American Heritage Series. (a) On the ...
Jemma's user avatar
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Conditional Probability Simulation Question

I have two variables $X$ and $Y$. $P(X<Y)=0$. I know $P(X=x)$ for all $x$ and $P(Y=y)$ for all $y$, how can I jointly simulate $10,000$ pairs of these values that fit these conditions? It seems ...
CamalotCoder's user avatar
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How to derive a negative binomial distribution from an uniform distribution?

I want to generate a negative binomial distribution from an uniform distribution. My attempt: U follows $Unif(0,1)$.$X =\lfloor ln(U) \rfloor$. $P(X=x)=P(\lfloor ln(U) \rfloor=x)$ =$P(x<=\lfloor ln(...
Faye4869's user avatar
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Probability of $x$ trials given $k$ successes

I am looking for a probability distribution that calculates the probability of $x$ total Bernoulli trials given fixed $k$ successes. I have looked into negative binomial distribution: $$ P(X=x) = \...
Marcus Chiu's user avatar
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Finding number n bernoulli trials (probability p) needed to satisfy k successes with probability j

Firstly sorry for the title. I know it's not a good title, but if I knew how to describe this problem in a more elegant way I likely wouldn't have it in the first place. The problem is the following: ...
Arelus's user avatar
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An infinite series expansion of $(1 + x)^n$ for all $n \in\mathbb{R}$?

I came across the following binomial identity which I had never previously encountered. $$(1+x)^n = \left(\frac{1}{1+x}\right)^{-n} = \left[\frac{1}{2}\left(1 + \frac{1-x}{1+x}\right) \right]^{-n} = ...
RTF's user avatar
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Is binomial distribution measuring combinations or successes? [closed]

So I think I understand what binomial coefficients are, "K number of combinations that can be made from set N". However, I keep seeing examples demonstrating K being the number of successes ...
InexperiencedCoder's user avatar
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Question on comparing two probability spaces

Let's assume we have two discrete probability spaces $(\Omega_1,P_1)$ which describes experiment $1$ and $(\Omega_2,P_2)$ which describes experiment $2$. We know the probability $P_1$ but we don't ...
Philipp's user avatar
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Question on solution of matchbox problem

Suppose a mathematician carries two matchboxes at all times: one in his left pocket and one in his right. Each time he needs a match, he is equally likely to take it from either pocket. Suppose he ...
Philipp's user avatar
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Question on applying negative binomial distribution

Let's assume we conduct an experiment which generates two outcomes: success $A$ or failure $B$. The experiment never generates more than $N$ failures. Both $A$ and $B$ have the same probability. After ...
Philipp's user avatar
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Probability space of Banach matchbox problem

Suppose a mathematician carries two matchboxes at all times: one in his left pocket and one in his right. Each time he needs a match, he is equally likely to take it from either pocket. Suppose he ...
Philipp's user avatar
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Is there a formulaic relation between the accuracy of the approximation and the number of terms used in the binomial expansion

So I am aware that with the increase in number of terms the approximation becomes more accurate but I wished to know if I have a binomial of the form say, $$ f(x) = (1-Rx)^{-n}$$ where $R$ is a real ...
Poke_Programmer's user avatar
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When is the binomial expansion with negative exponent valid for polynomials with complex numbers

So for a normal binomial expansion with negative exponent, say $$f(x) = (1 - 2x)^{-1},$$ we know it is valid for $$-1/2 < x < 1/2.$$ But what is the case for complex numbers? I was able to see ...
Poke_Programmer's user avatar
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Showing that $\mathbb{P}(X_1=k) = \frac{(1-p)^k}{k\log(1/p)}\quad k\in\mathbb{N}$ is a distribution on $\mathbb{N}$

Let $X_1, X_2,\ldots$ be independent and identically distributed random variables with the logarithmic mass function $$ \mathbb{P}(X_1=k) = \frac{(1-p)^k}{k \log(1/p)}, \quad k \in \mathbb{N} \quad (1)...
Landy's user avatar
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Why $\sum_{n=k}^\infty {n \choose k} p^{n-k} (1-p)^{k+1}=1$?

How can I see this: $$\sum_{n=k}^\infty {n \choose k} p^{n-k} (1-p)^{k+1}=1$$ What probability distribution that I can connect the above with? negative binomial?
Jackie's user avatar
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Estimating Negative Binomial parameters from subsampled data.

Suppose I need to estimate parameters $n$ and $p$ (or alternatively $\mu$ and $\sigma^2$) for a count data, that follows Negative Binomial distribution. However, I do not observe the raw counts, but ...
Dr. Timofey Prodanov's user avatar
2 votes
2 answers
118 views

A coin is tossed, coming up heads with probability $p$. What is the expected number of flips until there are $n$ heads total?

My thought process is: $X$ can only take on values greater than or equal to $n$. (Clearly, we cannot get $n$ heads in less than $n$ flips.) If $X=n$, that means every flip was a head, so this occurs ...
beeclu's user avatar
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Confusion about negative binomial theorem

It seems like that the order we choose influence the formula, but it shouldn't be. It should be like $(a+b)^{-n}=\sum_{k=0}^{\infty}\binom {-n}{k}a^k b^{-n-k}=\sum_{k=0}^{\infty}\binom {-n}{k}a^{-n-k}...
Zero's user avatar
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Negative binomial distribution — sum of two random variables with different success probabilities

Suppose $X, Y$ are independent random variables with $X\sim NB(r, p)$ and $Y\sim NB(s, q)$. From a previous post, I understand that when the success probabilities are equal, $p = q$, then, $$ X+Y \sim ...
Will's user avatar
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How long it takes to roll 6 6's in a row

Someone is sick and extremely bored. To pass the time, he decides to roll a $6$-sided die until he gets $6$ $6$'s in a row. If it takes him $5$ seconds for every roll of the die, how long is he ...
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Minimum Variance Unbiased Estimator for a Negative Binomial distribution

Given a negative binomial distribution parameterized by an unknown mean $\mu$ and known dispersion parameter $\alpha$ (so that the total variance is of the form $\mu + \alpha*\mu^2$), what is the ...
librus's user avatar
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Exponential matrix derivative to find the Hessian matrix of negative binomial regression

I am looking for the hessian matrix of the log likelihood function of negative binomial regression $$l\left( \cdot \right) =\sum ^{n}_{i=1}y_{i}\ln \left( \dfrac{\alpha \exp \left( x_{i}^{T}\beta \...
RFR Dina's user avatar
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Predict the probability of rooms being filled from a negative binomial process for arrival

This is somewhat of a riddle. I am trying to model the probability of filling up party rooms and simulation is not allowed. I know that the arrival of customers in 1 period of time (1 week) can be ...
cLwill's user avatar
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2 votes
2 answers
150 views

Compute $\mathbb{E}(X(X-1))$ where $X$ has a negative binomial distribution

Let's consider a Bernoulli trial where $p$ denotes the probability of success. A random variable $X$ that counts the frequency of failures until the $r$-th success has a negative binomial distribution....
Philipp's user avatar
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1 vote
2 answers
328 views

What is the probability that at least $11$ boxes must be purchased in order to obtain two of the prizes?

The probability that a randomly selected box of a certain type of cereal has a particular prize is $0.17$. Suppose that you purchase box after box until you have obtained $2$ of these prizes. (a) What ...
Olivia Manchenga's user avatar
1 vote
2 answers
156 views

Negative Binomial Distribution with finite number of trials

Suppose a user tosses a coin $n$ times. How do I compute the expected value of the number of heads before the user sees $k$ tails, ($k < n$)? This looks somewhat like Negative Binomial Distribution ...
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What is the probability of given data with the Negative binomial distribution?

A university exercise Statistics Learning and data analysis. This is the problem: Given that $ X=x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9, x_{10} = \left(2,3,7,8,2,4,7,5,5,7\right) $ What is the mean $ {\...
Georg's user avatar
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1 answer
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Writing the first-order loss function of the negative binomial distribution in terms of the CDF and the PMF.

So the negative binomial distribution uses two parameters $r$ and $p$, where $r \gt 0$ and $0 \lt p \lt 1$. We assume $r$ is an integer in this question. The probability mass function is defined as: $...
Steven01123581321's user avatar
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How many tails do you expect to see before getting 3 heads?

We have an unfair coin with a probability of 0.6 of obtaining a heads. Each flip is independent. How many tails do you expect to see before getting three heads? For example, imagine you see the ...
PleaseSir MayIHaveSomeMore's user avatar
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1 answer
120 views

Coupon collection problem and negative binomial distribution

The problem is: "Given 6 coupons, how many coupons do you expect you need to draw with replacement before having drawn each coupon at least once?" The solution is well-known: $$E[X] = E\left[...
Андрей Кокорев's user avatar
2 votes
2 answers
112 views

Generalizing the sum for the binomial expansion of a number with negative exponent

It is known that $$ \sum_{n=0}^{\infty} \binom{z+n-1}{n}(-1)^{n}x^{n} = (1+x)^{-z}, $$ but what about sums of the form $$ \sum_{n=t}^{\infty} \binom{z+n-1}{n}(-1)^{n}x^{n}? $$ Is there an explicit ...
Artur Wiadrowski's user avatar
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2 answers
94 views

Negative Binomial mean

Let $X\sim BN(r,p)$. If I try to compute E[X] through the Moment Generating Function I get the following: \begin{aligned} E[X] &=\left.\frac{d}{d t} M_{X}(t)\right|_{t=0} \ &=\left.\frac{d}{d ...
EmeViDji's user avatar
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Understanding binomial theorem with negative integer of n [duplicate]

I’m a student studying mathematics and I understood how the binomial theorem with positive integer of n works. But I couldn’t really understand the progress of how the binomial theorem with negative ...
Qwcmo's user avatar
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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 $20\%$. ...
August Jelemson's user avatar
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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 ...
TK99's user avatar
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1 answer
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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 $\frac{1}{x^2 + a^2 + 2ax}$? ?
aaksaksyk's user avatar
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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 ...
Dara's user avatar
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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 ...
InTheSearchForKnowledge's user avatar
2 votes
1 answer
62 views

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: $...
Mina's user avatar
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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-...
mielvil's user avatar
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1 vote
1 answer
326 views

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 ...
Monkey D. Erick's user avatar
4 votes
1 answer
124 views

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 $k\geqslant1$,...
user158293's user avatar
1 vote
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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{(...
user158293's user avatar
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1 answer
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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{...
frank's user avatar
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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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