Negative Binomial Distribution

As you know, Binomial distribution can give only two results, either it is a success or it is a failure. Let us understand it with an example- the tossing of the coin. You have to select one side, head, or tail. And, you will either, pass or fail. That is the Binomial Distribution. The negative binomial distribution is the same as the binomial distribution except for the fact that the events have a fixed number of trials. It is the discrete probability distribution for random variables in the negative binomial distribution. In this article, we are going to study the negative binomial Distribution, also some points about binomial Distribution, its difference from a negative binomial distribution, and the method for using a binomial distribution with discrete random variables.

Binomial Distribution

Binomial Distribution uses two parameters n and p. 

n gives you the number of times an experiment runs and p will be the probability of any one outcome. 

In binomial distribution, the number of success is determined by simply asking a question yes or no, Boolean value outcomes came in the form of true / success / yes or one ( Will be considered as probability p) and Boolean value of failure outcomes are failure / False / no / zero (taken as probability q, q = 1 – p). Bernoulli’s trial is a single test. 

Negative Binomial Distribution

In Probability and statistics, if a number of successes happen in series before a particularized failure, then this case will be known as a negative binomial distribution. Bernoulli trials should be independent and identically distributed. 

For a better understanding of negative binomial distribution, let us take an example, you have to throw a dice in which if you  2 on the dice then it will be considered a failure. And, all non 2’s are taken as success. 

Now, if you throw a dice until 2 appears for the third time. Then a number of failures, let’s say ‘r’, r = 3 failures. So, the probability distribution of non 2’s will be Negative Binomial Distribution in this case. 

Why it is called negative binomial?

A random variable is said to be the number of repeated trials that will give the success too. In other terms, it is said that it is the number of failures before any success. It is that difference that differentiates it from a normal binomial distribution. Here, in normal binomial distribution is calculated on the number of successes. When the word negative is used before binomial distribution, it comes to mind that the positive part has moved towards the negative side, making the distribution negative. So, in a binomial distribution, the number of successes will be counted while in a negative binomial distribution, the number of failures will be counted for an experiment. 

Some important points to remember in a negative binomial distribution-

• There must be x repeated trials in an experiment or event.

• There are two outcomes of the experiment- success or failure.

• The output of the trials is independent of each other. That means the output of one of them is independent of another. 

• The probability of getting success in an experiment will be the same in each and every trial. 

The formula for Negative Binomial Distribution

Probability Mass Function

For negative binomial Distribution, Probability Mass Function is, 

Where r is the number of failures and k is the number of successes. 

Properties

Expectation

For negative binomial Distribution, the expected number of successes with parameters (k,p) is k p/(1-p). 

In it, the average number of successes for an experiment is, N/n − k= k/(1 − p) – k = k p/(1 − p).

Variance

When we are counting the number of successes, The Variance for a given number of r of failures, the variance will be r p/(1 − p)2. 

When we count the number of failures before occurring of kth success, we get the variance is k(1-p) / p². 

Difference between Binomial Distribution and negative binomial Distribution

Binomial and negative binomial Distribution both works on the principle of the describes the distribution of “draws with replacement”. 

The main difference between them both is their stopping rule which is given below.

Binomial Distribution defines the k successes achieved in n trials which have a probability of success is p. 

While, negative binomial distribution describes the number of k successes,  until you get r failures, the probability of Success is p here also. 

So, the basic difference is that in binomial Distribution the number of trials is fixed but in negative binomial Distribution, the number of failures is fixed and the number of trials is random. 

For binomial Distribution and negative binomial Distribution, the without–replacement equivalents are hypergeometric and negative hypergeometric distributions respectively. 

Conclusion

We have read about, negative binomial Distribution in this article. Negative binomial Distribution means that, the probability distribution that is used with discrete random variables. This negative binomial distribution focuses on the number of trials that must occur before the occurrence of a defined number of successes. There is a little difference between the negative binomial distribution and binomial distribution as the former depends on the failures of the trial while the latter depends on the success of the trial.