Questions about the negative binomial distribution, a discrete probability distribution.
Questions tagged [negative-binomial]
273 questions
14
votes
2 answers
Random walk on a cube
Start a random walk on a vertex of a cube, with equal probability going along the three edges that you can see (to another vertex). what is the expected number of steps to reach the opposite vertex that you start with?
QRIUS2KNW
- 591
7
votes
4 answers
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 $i\in\{1,\dots,X\}$.
Finally, Let…
Um Shmum
- 449
6
votes
1 answer
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)}{p}
\end{align*}
Proof for 1.: Proof for the calculation…
punypaw
- 497
6
votes
4 answers
CDF for Negative Binomial Distribution
I am trying to show that the following statement is true.
$$
\sum_{x = r}^{X}\binom{x-1}{r-1}p^r(1-p)^{x-r} =
\sum_{x = r}^{X}\binom{X}{x}p^x(1-p)^{X-x}
$$
Where $X$ and $r$ and $p$ are constants, with $X \geq r$, and $ 0 \leq p \leq 1.$
How did I…
Neutrino
- 393
6
votes
2 answers
Find mean and variance using Moment generating function of the negative binomial.
I was asked to derive the mean and variance for the negative binomial using the moment generating function of the negative binomial.
However i am not sure how to go about using the formula to go out and actually solve for the mean and variance.
joe
- 99
5
votes
0 answers
Mathematics of Rock Paper Scissors game
The other day I was scrolling on my phone and came across the following meme:
For example, if 8 billion people had a Rock Paper Scissors tournament, apparently only 33 rounds would be needed to find the winner. I wrote a quick R script to verify…
farrow90
- 636
5
votes
1 answer
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 resulting in a black face?
Attemped Solution:
I'm trying…
Remy
- 8,244
5
votes
3 answers
Binomial expansion of negative exponents.
Let's say I have to expand $(1+x)^{-1}$ using binomial expansion.
Using the theorem, I get:
$$(1+x)^{-1} = 1-x+x^2-x^3+x^4-x^5+x^6+....+{\infty}$$
Substituting $x$ for $1$, I get:
$$\frac{1}{2}= 1-1+1-1+1-1+1+....+{\infty}$$
A similar result arises…
Chirag Arora
- 233
5
votes
1 answer
Probability Generating Function of a Negative Multinomial Distribution
Derive the probability generating function (pfg) of a negative multinomial distribution with parameters $(k; p_{0}, p_{1}, ..., p_{r})$ where the k-th occurrence of the event with the probability $p_{0}$ stops the trials.
My approach: Find the pgf…
Vectorizer
- 865
4
votes
1 answer
binomial identity seemingly illogical and impossible. Is there any way it could be true?
There is binomial expression(s) written as
$$\sum_{n\geqslant0}\frac{(-3n+2k-3)n!^2}{2(2n+1)(k-1)!^2(n-k+1)!^2 \binom{2n}{n}}=\begin{cases}
0 & \text{if $k=0$,} \\
-1 & \text{if $k\geqslant1$,}
\end{cases}$$
which simplifies…
user158293
- 488
4
votes
2 answers
How to split a pot of $100 when a game of flipping coins until first person gets 10 heads is interrupted.(A variation of the points problem)
Question :
Andy and Beth are playing a game worth $100. They take turns flipping
a penny. The first person to get 10 heads will win.
But they just realized that they have to be in math class right away
and are forced to stop the game.
Andy had four…
4
votes
3 answers
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 $\frac{1}{T-1}$ is an unbiased estimator of $p$.
My work so far:
I…
Kevin Frederick
- 129
4
votes
1 answer
Sum of two independent random variables ( negative binomial distribution )
Let $X,Y$ be two independent negative binomial distributed random variables: $X\sim NB(r,p)$ and $Y\sim NB(s,p)$. Show that:
$$X+Y\sim B(r+s,p)$$
Remark: So where I'm stucked?
I failed to show that $$ \sum_{j=0}^k\binom{j+r-1}j\cdot…
RukiaKuchiki
- 1,203
4
votes
1 answer
Partial sum of binomial expansion
Let $0m>0.$ I am looking for a closed form of
$\sum_{k=0}^{m}{n-k\choose m-k}p^k$ or $\sum_{k=0}^{m}{n-m+k\choose k}p^{m-k}$
4
votes
2 answers
Proving an inequality which the solution says "is obvious" but I cant see how
I was going through my Lebesgue Measure and Integration course, and I came across this inequality.
$$ \left(1+\frac{x}{n}\right)^{-n}\leq e^{-x} $$
I tried expanding both sides, and got
$$\text{LHS}=…
Sum-Meister
- 167
- 9