The Negative Binomial Distribution (Edexcel A Level Further Maths): 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,

  • 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  that has the negative binomial distribution you could write either:

    • or 

  • is the number of trials that will be required to reach a total of successes

  •  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 () 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 () is fixed

    • And you count the number of successes

  • While for a negative binomial distribution the number of successes () 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 th success

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

    • This also means the probability of success, , 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 , then  has the probability function:

  • 

  • the random variable is the number of trials needed to get  successes

  • is the constant probability of success in one trial

  • is the probability that the  success will occur on the trial

• Note that that is the product of

  • the binomial probability of getting successes in  trials,  ,

  • and the probability of getting a success in the  trial, 

• is the binomial coefficient

  • i.e., 

• Also note that there is no greatest possible value of

  • It could require any number of trials to reach the success

  • However for any given , gets closer and closer to zero as 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

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

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

  • is the probability of rolling a '6' on a fair dice

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

• The probability you are looking for is therefore 

• 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 , for 

    • 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 

  • So the probability of both those things happening is

• 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 ,the negative binomial probability function becomes:

• That is the same as the geometric distribution probability function

• The geometric distribution is the 'special case' of the negative binomial distribution when 

Negative Binomial Mean & Variance

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

• If , then

  • The mean of is 

  • The variance of is  

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