# Geometric distribution - Mathematics
Data Science for Political and Social Phenomena
Author: Guillaume Redoulès

# Geometric distribution

## Negative Binomial Experiment

A negative binomial experiment is a statistical experiment that has the following properties:

- The experiment consists of n repeated trials.
- The trials are independent.
- The outcome of each trial is either success (s) or failure (f).
- \(P(s)\) is the same for every trial.
- The experiment continues until x successes are observed

If \(X\) is the number of experiments until the \(x^{th}\) success occures, then \(X\) is a discrete random variable called a negative binomial

## Negative Binomial Distribution

Consider the following probability mass function:

$$b^*(x,n,p) = {\binom{n-1}{x-1}}p^xq^{n-x}$$

The function above is negative binomial and has the following properties:

- The number of successes to be observed is \(x\).
- The total number of trials is \(n\).
- The probability of success of 1 trial is \(p\).
- The probability of failure of 1 trial \(q\), where \(q=1-p\).
- \(b^*(x,n,p)\) is the negative binomial probability, meaning the probability of having exactly \(x-1\) successes out of \(n-1\) trials and having \(x\) successes after \(n\) trials.

## Geometric Distribution

The geometric distribution is a special case of the negative binomial distribution that deals with the number of Bernoulli trials required to get a success (i.e., counting the number of failures before the first success). Recall that \(X\) is the number of successes in \(n\) independent Bernoulli trials, so for each \(i\) (where $1\leq i\leq n):

$ X_i =

\begin{cases} 1 if the i^{th} trial is a success \\ 0 otherwise x \end{cases}

$

The geometric distribution is a negative binomial distribution where the number of successes is 1. We express this with the following formula:

$$g(n,p)=q^{n-1}p$$

### Example

Bob is a high school basketball player. He is a 70% free throw shooter, meaning his probability of making a free throw is 0.7. What is the probability that Bob makes his first free throw on his fifth shot?

For this experiment n=5, p=0.7 and q=0.3 So :

$$g(n=5, p=0.7)=0.3^4 0.7=0.00567$$

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