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April 13, 2023

# Geometric Distribution - Definition, Formula, Mean, Examples

Probability theory is an essential division of mathematics which deals with the study of random events. One of the essential theories in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution which models the number of tests required to get the first success in a sequence of Bernoulli trials. In this article, we will explain the geometric distribution, derive its formula, discuss its mean, and provide examples.

## Meaning of Geometric Distribution

The geometric distribution is a discrete probability distribution which describes the number of experiments needed to accomplish the first success in a sequence of Bernoulli trials. A Bernoulli trial is an experiment which has two viable results, typically indicated to as success and failure. Such as flipping a coin is a Bernoulli trial since it can likewise come up heads (success) or tails (failure).

The geometric distribution is used when the trials are independent, meaning that the consequence of one test does not impact the result of the next trial. In addition, the chances of success remains unchanged across all the trials. We could denote the probability of success as p, where 0 < p < 1. The probability of failure is then 1-p.

## Formula for Geometric Distribution

The probability mass function (PMF) of the geometric distribution is specified by the formula:

P(X = k) = (1 - p)^(k-1) * p

Where X is the random variable that depicts the amount of trials needed to achieve the first success, k is the count of trials required to achieve the initial success, p is the probability of success in a single Bernoulli trial, and 1-p is the probability of failure.

Mean of Geometric Distribution:

The mean of the geometric distribution is described as the likely value of the number of experiments required to achieve the first success. The mean is given by the formula:

μ = 1/p

Where μ is the mean and p is the probability of success in an individual Bernoulli trial.

The mean is the anticipated count of trials needed to get the first success. For example, if the probability of success is 0.5, therefore we expect to obtain the first success following two trials on average.

## Examples of Geometric Distribution

Here are few primary examples of geometric distribution

Example 1: Flipping a fair coin up until the first head appears.

Suppose we flip an honest coin till the initial head appears. The probability of success (obtaining a head) is 0.5, and the probability of failure (getting a tail) is also 0.5. Let X be the random variable that portrays the count of coin flips required to obtain the initial head. The PMF of X is given by:

P(X = k) = (1 - 0.5)^(k-1) * 0.5 = 0.5^(k-1) * 0.5

For k = 1, the probability of obtaining the first head on the first flip is:

P(X = 1) = 0.5^(1-1) * 0.5 = 0.5

For k = 2, the probability of getting the first head on the second flip is:

P(X = 2) = 0.5^(2-1) * 0.5 = 0.25

For k = 3, the probability of achieving the first head on the third flip is:

P(X = 3) = 0.5^(3-1) * 0.5 = 0.125

And so on.

Example 2: Rolling an honest die up until the initial six appears.

Suppose we roll a fair die till the initial six appears. The probability of success (getting a six) is 1/6, and the probability of failure (achieving any other number) is 5/6. Let X be the irregular variable which portrays the number of die rolls required to obtain the first six. The PMF of X is given by:

P(X = k) = (1 - 1/6)^(k-1) * 1/6 = (5/6)^(k-1) * 1/6

For k = 1, the probability of getting the first six on the first roll is:

P(X = 1) = (5/6)^(1-1) * 1/6 = 1/6

For k = 2, the probability of getting the initial six on the second roll is:

P(X = 2) = (5/6)^(2-1) * 1/6 = (5/6) * 1/6

For k = 3, the probability of obtaining the first six on the third roll is:

P(X = 3) = (5/6)^(3-1) * 1/6 = (5/6)^2 * 1/6

And so on.

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The geometric distribution is an essential concept in probability theory. It is applied to model a wide range of practical phenomena, such as the count of trials needed to achieve the first success in different scenarios.

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