The Negative Binomial Distribution (Edexcel A Level Further Maths: Further Statistics 1): Revision Note

Conditions for negative binomial models

What is the negative binomial distribution?

• The negative binomial distribution models the number of trials needed to reach a fixed number of successes, $r$

  • For example, how many times will you have to roll a dice until it lands on a '6' for the third time

•  There is no one standard form of notation for the negative binomial distribution

  • But for a random variable $X$ that has the negative binomial distribution you could write either:

    • $X~\mathrm{NB}\left(r,p\right)$ or $X~\mathrm{Negative}\mathrm{B}\left(r,p\right)$

  • $X$ is the number of trials that will be required to reach a total of $\mathit{r}$ successes

  • $p$ is the fixed probability of success in any one trial

What are the conditions for using a negative binomial model?

• A negative binomial model can be used for an experiment that satisfies the following conditions:

  • The experiment consists of an indefinite number of successive trials

  • The outcome of each trial is independent of the outcomes of all other trials

  • There are exactly two possible outcomes for each trial (success and failure)

  • The probability of success in any one trial ($p$) is constant 

•  Note that these conditions are very similar to the conditions for the binomial distribution

  • But for a binomial distribution the number of trials ($\mathit{n}$) is fixed

    • And you count the number of successes

  • While for a negative binomial distribution the number of successes ($\mathit{r}$) is fixed

    • And you count the number of trials it takes to reach that number of successes

When might the conditions not be satisfied?

• If asked to criticise a negative binomial model, you may be able to question whether the trials are really independent

  • For example, someone may be repeating an activity until they achieve the $r$th success

    • The trials may not be independent because the person gets better from practising the activity

    • This also means the probability of success, $p$, is not constant

  • In order to proceed using the model, you will have to assume trials are independent

Negative binomial probabilities

What are the probabilities for the negative binomial distribution?

• If $X~\mathrm{Negative}\mathrm{B}\left(r,p\right)$, then $X$ has the probability function:

  • $\mathrm{P}\left(X=x\right)=\mathrm{p}\left(x\right)\mathit{=}\mathit{\left(}\begin{array}{c}x-1\\ r-1\end{array}\mathit{\right)}{p}^r{\mathit{(}1\mathit{-}p\mathit{)}}^{x\mathit{-}r}\mathit{,}x=r\mathit{,}r+1,r+2,...$

  • the random variable $X$ is the number of trials needed to get $\mathit{r}$ successes

  • $p$ is the constant probability of success in one trial

  • $\mathrm{P}\left(X=x\right)$ is the probability that the $r\mathrm{th}$ success will occur on the $x\mathrm{th}$ trial

• Note that that is the product of

  • the binomial probability of getting $\mathit{r}\mathbf{-}\mathbf{1}$ successes in $\mathit{x}\mathbf{-}\mathbf{1}$ trials,  $\mathit{\left(}\begin{array}{c}x\mathit{-}\mathit{1}\\ r\mathit{-}\mathit{1}\end{array}\mathit{\right)}{p}^{r-1}{\mathit{(}\mathit{1}\mathit{-}p\mathit{)}}^{x\mathit{-}r}$,

  • and the probability of getting a success in the $\mathit{x}\mathbf{th}$ trial, $p$

• $\left(\begin{array}{c}x-1\\ r-1\end{array}\right)$ is the binomial coefficient

  • i.e., $\left(\begin{array}{c}x-1\\ r-1\end{array}\right)={}^{\left(x-1\right)}C_{\left(r-1\right)}$

• Also note that there is no greatest possible value of $x$

  • It could require any number of trials to reach the $r\mathrm{th}$ success

  • However for any given $r$, $\mathrm{P}\left(X=x\right)$ gets closer and closer to zero as $x$ gets larger

Where does the formula come from?

• Consider rolling a fair dice and wanting to know the probability that it would take 12 rolls for the dice to land on '6' a total of 3 times

  • This is the same as saying that the third '6' occurs on the 12th roll

• You can model this situation using the random variable $X~\mathrm{Negative}\mathrm{B}\left(3,\frac{1}{6}\right)$

  • 'success' here is defined as 'roll a 6'

  • $r=3$ because we're interested in the number of trials required to reach a total of 3 successes

  • $p=\frac{1}{6}$ is the probability of rolling a '6' on a fair dice

  • $X$ is the number of trials that will be required to reach 3 successes

• The probability you are looking for is therefore $\mathrm{P}\left(X=12\right)$

• For the third '6' to occur on the 12th roll, the following things need to happen:

  • '6' must occur exactly 2 times in the first 11 rolls

    • It doesn't matter which rolls those two '6's occur on

    • The probability of that happening is $\mathrm{P}\left(Y=2\right)$, for $Y~\mathrm{B}\left(11,\frac{1}{6}\right)$

    • So that part of the answer is a binomial probability

  • Then a '6' must also occur on the 12th roll

    • The probability of that happening is $\frac{1}{6}$

  • So the probability of both those things happening is

$\mathrm{P}\left(Y=2\right)\times \frac{1}{6}=\left(\left(\begin{array}{c}11\\ 2\end{array}\right){\left(\frac{1}{6}\right)}^2{\left(\frac{5}{6}\right)}^9\right)\times \frac{1}{6}=\left(\begin{array}{c}11\\ 2\end{array}\right){\left(\frac{1}{6}\right)}^3{\left(\frac{5}{6}\right)}^9=0.0493489...$

• That same logic can be used to find the general formula for negative binomial probabilities

Is there a connection between negative binomial probabilities and geometric probabilities?

• Note that when $r=1$,the negative binomial probability function becomes:

$\mathrm{P}\left(X=x\right)=\mathrm{p}\left(x\right)\mathit{=}\mathit{\left(}\begin{array}{c}x\mathit{-}1\\ 0\end{array}\mathit{\right)}{p}^1{\mathit{(}1\mathit{-}p\mathit{)}}^{x\mathit{-}1}=p{\left(1-p\right)}^{\mathit{x}\mathit{-}\mathit{1}}\mathit{,}x=1\mathit{,}2,3,...$

• That is the same as the geometric distribution probability function

• The geometric distribution is the 'special case' of the negative binomial distribution when $r=1$

Negative binomial mean & variance

What are the mean and variance of the negative binomial distribution?

• If $X~\mathrm{Negative}\mathrm{B}\left(r,p\right)$, then

  • The mean of $X$ is  $\mathrm{E}\left(X\right)=\mu \mathit{=}\frac{\mathit{r}}{\mathit{p}}$

  • The variance of $X$ is  $\mathrm{Var}\left(X\right)={\sigma }^{\mathit{2}}\mathit{=}\frac{\mathit{r}\left(1-p\right)}{{\mathit{p}}^{\mathit{2}}}$

• You need to be able to use these formulae to answer questions about the negative binomial distribution.