# Geometric Distribution - Definition, Formula, Mean, Examples

Probability theory is an essential division of mathematics that 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 number of tests needed to obtain the first success in a series of Bernoulli trials. In this article, we will talk about the geometric distribution, derive its formula, discuss its mean, and provide examples.

## Explanation of Geometric Distribution

The geometric distribution is a discrete probability distribution which narrates the number of trials required to achieve the initial success in a series of Bernoulli trials. A Bernoulli trial is a trial that has two possible results, typically indicated to as success and failure. For instance, flipping a coin is a Bernoulli trial because it can likewise come up heads (success) or tails (failure).

The geometric distribution is utilized when the trials are independent, meaning that the result of one trial doesn’t affect the outcome of the upcoming test. Furthermore, the chances of success remains constant throughout all the trials. We can indicate 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 which depicts the amount of test needed to attain the initial success, k is the count of tests needed to attain 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 defined as the anticipated value of the amount of test required to obtain 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 expected count of experiments required to obtain the initial success. Such as if the probability of success is 0.5, therefore we expect to obtain the initial success after 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.

Imagine we toss an honest coin till the first head appears. The probability of success (getting a head) is 0.5, and the probability of failure (obtaining a tail) is also 0.5. Let X be the random variable which depicts the number of coin flips needed 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 initial head on the first flip is:

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

For k = 2, the probability of achieving 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 getting 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 a fair die until the initial six appears.

Let’s assume we roll a fair die up until the initial six shows up. The probability of success (achieving a six) is 1/6, and the probability of failure (getting all other number) is 5/6. Let X be the random variable which portrays the count of die rolls needed to obtain the initial six. The PMF of X is stated as:

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 initial roll is:

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

For k = 2, the probability of getting the first 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 forth.

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The geometric distribution is an essential theory in probability theory. It is used to model a broad array of practical phenomena, for instance the number of tests needed to get the initial success in different scenarios.

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