Questions tagged [negative-binomial]

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

Filter by
Sorted by
Tagged with
0
votes
0answers
25 views

Poisson alternatives [closed]

As a part of my Mathematics course, I am writing a paper on the probability of my favourite Youtuber's video uploads. I applied the Poisson distribution for the upload period of two years, where I ...
1
vote
1answer
27 views

How to taylor expand $1/(1- \frac{k}{r})$ where k is some constant

Hi i am doing some integrations and there came across one step that i couldn't understand. The step is as follows: $$\frac{dr}{1 - \displaystyle\frac{k}{r}}=dr\left(1 + \frac{k}{r-k}\right)$$. I don't ...
-1
votes
1answer
21 views

Negative Multinomial Coefficients [closed]

In case of negative binomial coefficients ,they were used to count number of solutions of an linear equation with n variable or distribute r things among n people. So my question is could anyone ...
1
vote
0answers
22 views

An electronic scale in an automated filling operation stops the manufacturing line after five overweight packages are detected.

An electronic scale in an automated filling operation stops the manufacturing line after five overweight packages are detected. Suppose that the probability of an overweight package is 0.05 and each ...
0
votes
2answers
40 views

How was this simplification done?

\begin{align*} \begin{pmatrix} -2\\ n\end{pmatrix}&=\frac{(-2)(-3)(-4)\dots(-2-n+1)}{n!}\\[5pt] &= (-1)^n\frac{2\times3\times4\times\dots n(n+1)}{n!}\\[5pt] &=(-1)^n(n+1)\\[...
0
votes
0answers
20 views

Conditional variance of random variables sum

I am trying to resolve the probability exercise but I still cannot get the proper result. $N, Z_{1}, Z_{2}, ..., Z_{n}, ... $ and $(X_{1}, Y_{1}), (X_{2}, Y_{2}), ..., (X_{n}, Y_{n}), ...$ are ...
0
votes
0answers
7 views

quasiconcavity of a truncated and scaled negative binomial expectation

Let $P_1\triangleq\bar{F}(\frac{p_1}{a_1})$ and $P_2\triangleq\bar{F}(\frac{p_2}{a_2})$ where $F(\cdot)$ is c.d.f. with Lebesgue p.d.f. $f(\cdot)$. We want to solve following optimization problem \...
1
vote
2answers
57 views

How to evaluate $ \sum_{n=r}^{\infty} n^2 \binom{n-1}{r-1} p^r (1-p)^{n-r}$?

I want to evaluate the following sum $$ \sum_{n=r}^{\infty} n^2 \binom{n-1}{r-1} p^r (1-p)^{n-r} $$ where $r \in \mathbb{N}$ and $0<p<1$. Using WA I know that the sum evaluates to $\frac{r(r-...
0
votes
1answer
27 views

How to compare binomial distributions with different number of trial?

I want to know which event has the best score (or luck or odd) $p$ (prob of success) is equal with all event. There are givne set of events, that is (number of try, number of success) $$\mathrm{event1}...
2
votes
0answers
16 views

Negative binomial (and multinomial) extension to the minimum(failures,successes)

The negative binomial distribution defines the probability of N draws to obtain K successes. The negative multinomial gives the joint over k1...km given k0 successes. I need to find the PMF over n ...
0
votes
0answers
17 views

Lower index becoming negative upon solving

The question While solving this question by using the property: r nCr = n n-1Cr-1 The value of lower index becomes negative which is invalid How do I proceed?
0
votes
0answers
7 views

Marginal Dirichlet Negative Binomial Distribution and the Multinomial Inverse Polya Urn

I have the following 'URN-like' problem - assume an urn the contains balls with m different colors. As in the standard Polya scheme, every time a ball is sampled, it is returned the urn in addition ...
0
votes
2answers
43 views

Negative Binomial Series

On this webpage, the expansion of the negative binomial series is given below. $$ (x+a)^{-n} = \sum^\infty_{k=0} (-1)^k \begin{pmatrix}n + k - 1\\k\end{pmatrix} x^ka^{-n-k} $$ when $|x| < a$. My ...
0
votes
0answers
21 views

Hypothesis testing with the weighted average of negative binomial distributions

I am trying to do a statistical hypothesis testing between two random variables $Y^{(1)}$ and $Y^{(2)}$: $Y^{(1)} = \frac{\sum_{i=1}^{n}w_i^{(1)}X_i^{(1)}}{\sum_{i=1}^{n}X_i^{(1)}}$ $Y^{(2)} = \frac{\...
1
vote
0answers
43 views

Uniformly minimum variance unbiased estimator for negative binomial distribution

I'm working through an old qualifying exam and I feel like I'm so close to the answer. Given a random sample of size n from a negative binomial distribution with parameters (r,p) I need to find a UMVU ...
-2
votes
1answer
177 views

Minimum Variance Estimators that Attain the Cramér-Rao lower bound

The probability mass function of a negative binomial random variable $X$ is given by $$P(X=x\mid\theta)= {r+x-1 \choose x}(1-\theta)^x\theta^r$$ and $\Bbb E(X\mid \theta)=\frac{r(1-\theta)}{\theta}$. ...
-1
votes
1answer
40 views

E[1/(Y+1)] of negative binomial distribution [closed]

There was an exercise in class asking to solve for the expectation E[1/(Y+1)] when Y is a negative binomial random variable (number of failures). How do I solve this? Whatever transformation or ...
1
vote
1answer
57 views

is $\frac{Γ(r+x)}{Γ(r)Γ(x+1) }$ the same as ${r+x-1 \choose x}$?

is $\frac{Γ(r+x)}{Γ(r)Γ(x+1) }$ the same as ${r+x-1 \choose x}$ ? I have two pmfs $$P(X=x\mid\theta)={r+x-1 \choose x}(1-\theta)^x\theta^r $$ and $$P(X=x\mid\theta)=\frac{Γ(r+x)}{Γ(r)Γ(x+1) }(1-\theta)...
0
votes
1answer
21 views

Quick question about the definition of a random variable in a baseball pitches example

I'm just wondering, in the last sentence of the third paragraph, why can X not be larger than 6? Is that because the number of innings is 9? I'm really confused, I thought X goes from 3,4,5,6,...to ...
0
votes
0answers
18 views

Probability of having s success after $(s-n)$ failure in a coin toss (Bernoulli Experiment)

I have a interesting problem: Two people take turns tossing a coin. If the coin lands on heads, person $A$ gains one point, and person $B$ loses one point. If the coin lands on tails, person $A$ loses ...
0
votes
1answer
185 views

Expectation of negative binomial distribution

Given $X \sim \text{NBin}(n,p)$, I've seen two different calculations for $\mathbb{E} (X)$: \begin{align*} &1. \mathbb{E} (X) = \frac{n}{p}, \quad \text{or}\\ &2. \mathbb{E} (Y) = \frac{n(1-p)}...
0
votes
0answers
42 views

Finding the expectation $\mathrm{E} (1/ X)$ for a negative binomial random variable $X$.

Suppose a random variable $X$ is distributed as $\operatorname{NB}(\mu, \theta)$, and its mass is as follows $$ \mathrm{P}(X = y) = \binom{y + \theta - 1}{y} \left(\frac{\mu}{\mu + \theta}\right)^{y}\...
0
votes
1answer
28 views

What is the size of a multinomial?

The answer to this question uses the phrase "multinomial of size". What is the definition of the size of a multinomial? They are using a negative multinomial.
0
votes
1answer
84 views

PGF of negative multinomial expansion

I have found the formula for the Probability Generating Function of negative multinomial distribution in Definition 8.1 of this chapter (https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118445112....
-1
votes
3answers
60 views

When rolling a die, what is the probability that a 4 has appeared at least 3 times by the 15th roll?

Now I've managed to figure out the probability that the 3rd time a 4 appears is on the 15th roll using a negative binomial distribution. But the at least is totally screwing with me in this variation ...
1
vote
0answers
50 views

Negative multinomial expansion

I would like to know the standard form of the negative multinomial expansion i.e. $(x_1 + x_2 + \ldots + x_p)^{-n}$. I understand that I can probably derive something by applying the negative binomial ...
1
vote
1answer
49 views

Negative Binomial Distribution MGF

The Negative Binomial Distribution Wikipedia page lists its Moment-Generation Function as $(\frac{1 - p}{1 - pe^t})^r$. However, the Moment-Generating Function page lists the MGF of the Negative ...
2
votes
2answers
83 views

Find the coefficient of ${x}^{20 }$ in ${({x}^{2}+{x}^{3}+{x}^{4}+{x}^{5}+{x}^{6})}^{5}$

I saw a question in my textbook, the solution of this question exists in my textbook. However , its solution is very long.I tried to solve it in different way but i do not know whether it is true or ...
0
votes
1answer
59 views

Why is the negative binomial distribution the sum of r geometric distributions?

Does anyone understand why the negative binomial distribution is the sum of r geometric distributions? See the example given below. Could someone please explain the logic behind this statement and ...
0
votes
0answers
36 views

How to deduce that a discrete random variable $X$ is bimodal?

The following question is on negative binomial distribution. Does anyone have any clues on how to solve part (b)(ii) (see below the pink stripes in the picture)? Any explanation or tips are welcome.
0
votes
3answers
29 views

How express numerator of: Given that after 10 days Ian has had 8 invitations, find the P that he will receive the 9th invitation on the 14th day? [closed]

Below is the question that I have been working with and I am stuck on part (c). I approached this question by working with conditional probability, and hence the formula P(A given B)=P(A and B)/P(B), ...
1
vote
2answers
131 views

Fibonacci general term: How do I mend this broken derivation?

I am working on a derivation of the general term for the Fibonacci Sequence. Here's where I want to be: $$F(n)=\frac{1}{\sqrt5}(\phi^n-\overline{\phi}^n)$$ Here, $F(n)$ is the $n^\text{th}$ term, $\...
0
votes
0answers
12 views

How to solve negative binomial regression

From this https://content.wolfram.com/uploads/sites/19/2013/04/Zwilling.pdf : But I don't know how to I know let the derivative of the function L with respect to $\alpha$ and $\beta$ equal to zero , ...
0
votes
2answers
58 views

Bounding coefficients of the polynomial $(1-x^2)^n (1-x)^{-m}$.

I have been trying to get some upper bound on the coefficient of $x^k$ in the polynomial $$(1-x^2)^n (1-x)^{-m}, \text{ $m \le n$}.$$ A straightforward calculation shows that for even $k$, the ...
1
vote
0answers
14 views

Probability of gaining a game series

Let's say I have a team, and we are playing a game against another team, where our probability of winning each game is $p$. We get the offer to play either $2k-1$ games or $2k+1$ games, where $k$ is ...
0
votes
2answers
28 views

Which one is the correct mean formula of negative binomial distribution? $\frac{r}{p}$ or $\frac{(1-p) r}{p}$?

If we are looking to find the probability of observing the $6th$ head after $12$ independent flips and we let $X$ be the random variable for the number of flips of an unbiased coin I found that there ...
1
vote
2answers
102 views

To find the finite summation of negative binomial.

I'm looking to simplify the following expression. Basically need to remove the summation up to $t$ $$\sum_{k=j}^t \binom{k-1}{j-1} p^{k-j} q^j$$ where $p=1-q$, $t$ is a large finite number like $10^{...
-2
votes
2answers
39 views

Making a Negative Number Possible to Square Root

We are able to solve $x^2+4=0$ by square rooting both sides, but if we have $x^2=-4$ we can't solve. Firstly, why? Aren't they equal expressions? Secondly, if we have $x^2=-4$, why can't we bring the ...
0
votes
0answers
7 views

Negative binomial Confusion

While studying I came across with a problem that when the observable phenomenon give rise to the discrete distributions which show a variance larger than the mean. We use negative binomial ...
0
votes
0answers
53 views

Negative binomial distribution, conditional probability, and proving the identity

Let $X$ and $Y$ have a common negative binomial distribution. Find the conditional probability $P(X= j | X+Y = k )$ and show the identity $$\sum_{j=0}^k {a+k -j -1 \choose k-j} {b+j -1 \choose j} = ...
0
votes
2answers
76 views

Is my formula for the CDF of negative binomial distribution right?

Due to the differences in notation for the formula of the CDF of negative binomial distribution from Wikipedia, ScienceDirect and Vose Software, I decide to rewrite it in the way that I can easily ...
0
votes
1answer
76 views

Relationship between binomial and negative binomial probabilities

Let $X$ be a negative binomial random variable with parameters $r$ and $p$, and let $Y$ be a binomial random variable with parameters $n$ and $p$. Show that $$ \mathbb{P}(X > n) = \mathbb{P}(Y < ...
0
votes
1answer
38 views

Poker cards probability distribution

A card is drawn from a poker deck, inspected and put back into the deck. This process is repeated until 2 aces are obtained. What is the probability that this will be achieved in less than 6 attempts? ...
2
votes
3answers
39 views

Negative binomial distribution and negative binomial series missing $(-1)^k$ term

The pmf of a negative binomial distribution is $$p_X(x)= {x-1 \choose r-1}~ p^r~ (1-p)^{x-r}\quad x=r,r+1,\cdots$$ I want to verify that $$\sum \limits_{x=r}^{\infty} p_X(x)= 1$$ I start with $$\...
0
votes
0answers
8 views

How to find parameter k of negative binomial distribution using size and mu data?

I fit a negative binomial distribution and I have size = 0.47 and mu = 134. 85 for population 1, and size = 0.26 and mu = 136.49 in population 2. How do I find the parameter k, to infer which of the ...
2
votes
1answer
181 views

Probability of winning a coin flipping game

Let's say two people are playing a game, where each one flips a coin with some unknown probability of success (e.g. a heads), given by $p_1$ and $p_2$. A player wins the game when they get to 21 heads....
1
vote
1answer
38 views

summation of series (binomial theorem)

I need help in solving this question. If $n \in N$,$n\ge 3$ then the value of $(1)n-\frac{(n-1)}{1!}+\frac{(n-1)(n-2)^2}{2!}-\frac{(n-1)(n-2)(n-3)^2}{3!}+.....$ upto n terms any help hint will do. ...
0
votes
0answers
15 views

Prove Pascal distribution by Taylor series

I attempt to prove it by my analysis knowledge instead of the independence. Let $\mathit{T}_r$ denote the number of trials until the $r^{th}$ success in Bernoulli (p) trials. $$\Pr(\mathit{T}_r = t) = ...
0
votes
0answers
42 views

Find $\lim\limits_{n \to +\infty} np^n\sum\limits_{k=n}^{\infty} \frac{1}{k} \binom{k-1}{n-1}(1-p)^{k-n}$. [duplicate]

This is very likely to be a probabilitic problem,just noting that $$E[1/X]=\sum_{k=n}^{\infty} \frac{1}{k}\binom{k-1}{n-1}p^n(1-p)^{k-n},$$ where $X$ is the random variable of Negative-Binomial ...
0
votes
0answers
26 views

Generalized negative binomial distribution

In "Conditional Specification of Statistical Models", Arnold, $et$ $al.$, 1999, following generalization of negative binomial distribution is presented (p. 98) : $$ f(x) = \frac{\Gamma(m_{02})}{{(m_{...

1
2 3 4 5