## Introduction

Negative Binomial Distribution (also known as Pascal Distribution) should satisfy the following conditions;

- The experiment should consist of a sequence of independent trials.
- Each trial should have only 2 outcomes. That is Success (S) or Failure (F).
- Success Probability θ should be constant from trial to trial.
- The experiment should be continued until the occurrence of r total successes. Here r is a specified positive integer.

In the Binomial Distribution, we were interested in the number of **Successes **in n number of trials. But in the Negative Binomial Distribution, we are interested in the number of **Failures** in n number of trials. This is why the prefix “Negative” is there.

## Differences between Binomial Random Variable and Negative Binomial Random Variable

Negative Binomial Random Variable | Negative Binomial Random Variable |
---|---|

Fixed No. of trials. | Trials are not fixed, random. |

No. of successes is not fixed. | No. of failures are fixed. |

## Detailed Example – 1

Consider an experiment where we roll a die until the face 6 turns upwards **two** times.

Here;

- Our trials are independent. That means turning 6 face upwards on one trial does not affect whether or 6 face turns upwards on the next trials.
- Each trial only has 2 outcomes.
- Success probability is constant. 1/6 for every trial.
- The experiment is continued until the 6 face turns upwards 2 times.

## Probability Distribution

#### X = Number of failures that precede the **r**^{th} success.

^{th}

(1)

**Terminology;**

- x – value of the Random Variable X
- r – r
^{th}success. (no. of failures = (r-1)) - θ – Success probability

## Detailed Example – 2

A researcher is interested in examining the relationship between students’ mental health and their exam marks. For this, he wishes to conduct interviews with 5 students.

θ = Probability of a randomly selected student agrees to sit for the interview

θ = 0.2

What is the probability that 15 students should be asked before 5 students are found to agree to sit for the interview?

(2)

## Expected Value and Variance

Expected number of trials until first success is;

(3)

Therefore, expected number of failures until first success is;

(4)

Hence, we expect failures before the r^{th} success.

Therefore,

(5)

## Geometric Distribution

This is a special case of Negative Binomial Distribution where **r=1**

That means, we are interested in finding number of trials that is required for a **single success.**

**Example :**

Tossing a coin until it lands on heads.

#### Formula

(6)

##### References

- Probability and Statistics for Engineering and the Sciences 8th Edition by Jay. L. Devore
- https://stattrek.com/ – This site features very high quality statistics content!