# Geometric Distribution - Definition, Formula, Mean, Examples

Probability theory is an essential department of math that takes up the study of random occurrence. One of the important concepts in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution which models the amount of trials required to get the first success in a sequence of Bernoulli trials. In this article, we will define the geometric distribution, derive its formula, discuss its mean, and offer examples.

## Definition of Geometric Distribution

The geometric distribution is a discrete probability distribution that describes the number of experiments required to achieve the initial 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 flipping a coin is a Bernoulli trial because it can either come up heads (success) or tails (failure).

The geometric distribution is used when the experiments are independent, which means that the consequence of one test doesn’t impact the result of the next trial. Furthermore, the probability of success remains same across 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 given by the formula:

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

Where X is the random variable which portrays the number of trials required to get the first success, k is the number of tests required to attain the initial 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 described 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 an individual Bernoulli trial.

The mean is the anticipated number of trials needed to get the initial success. For example, if the probability of success is 0.5, then we anticipate to attain the initial 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 shows up.

Suppose we flip a fair coin until the initial head turns up. The probability of success (obtaining a head) is 0.5, and the probability of failure (obtaining a tail) is also 0.5. Let X be the random variable which portrays the count of coin flips required to get the initial head. The PMF of X is stated 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 obtaining 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 obtaining the initial head on the third flip is:

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

And so forth.

Example 2: Rolling a fair die till the first six appears.

Let’s assume we roll an honest die until the initial six appears. 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 irregular variable which depicts the number of die rolls required to achieve 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 first roll is:

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

For k = 2, the probability of obtaining 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 getting 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 a crucial theory in probability theory. It is used to model a broad range of real-world phenomena, for instance the count of tests needed to get the initial success in various situations.

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