# Geometric Distribution - Definition, Formula, Mean, Examples

Probability theory is an essential department of mathematics which takes up the study of random events. One of the crucial concepts in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution which models the amount of trials needed to obtain the first success in a series of Bernoulli trials. In this article, we will define the geometric distribution, derive its formula, discuss its mean, and provide examples.

## Explanation of Geometric Distribution

The geometric distribution is a discrete probability distribution that portrays the amount of experiments required to reach the first success in a sequence of Bernoulli trials. A Bernoulli trial is a trial that has two possible results, generally indicated to as success and failure. Such as tossing a coin is a Bernoulli trial since it can either come up heads (success) or tails (failure).

The geometric distribution is used when the experiments are independent, which means that the outcome of one test doesn’t affect the outcome of the next test. Additionally, the probability of success remains same across all the tests. 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 given by the formula:

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

Where X is the random variable which depicts the number of trials required to get the initial success, k is the number of experiments required to achieve the first success, p is the probability of success in an individual Bernoulli trial, and 1-p is the probability of failure.

Mean of Geometric Distribution:

The mean of the geometric distribution is defined as the anticipated value of the number of test needed to obtain the initial success. The mean is given by the formula:

μ = 1/p

Where μ is the mean and p is the probability of success in a single Bernoulli trial.

The mean is the expected count of experiments needed to obtain the first success. Such as if the probability of success is 0.5, then we expect to attain the first success after two trials on average.

## Examples of Geometric Distribution

Here are some basic examples of geometric distribution

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

Imagine we flip a fair coin until the first 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 first head. The PMF of X is provided as:

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

For k = 1, the probability of getting 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 obtaining the initial head on the second flip is:

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

For k = 3, the probability of obtaining the initial head on the third flip is:

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

And so on.

Example 2: Rolling a fair die up until the initial six shows up.

Suppose we roll an honest die until the initial six appears. The probability of success (getting a six) is 1/6, and the probability of failure (obtaining all other number) is 5/6. Let X be the irregular variable which depicts the count of die rolls required to get 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 obtaining the initial six on the initial 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 getting the initial 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 theory in probability theory. It is applied to model a wide range of real-world scenario, for instance the count of experiments needed to get the first success in different scenarios.

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