# Geometric Distribution - Definition, Formula, Mean, Examples

Probability theory is ac crucial department of math that takes up the study of random events. One of the essential ideas in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution that models the number of trials required to obtain the initial success in a series of Bernoulli trials. In this article, we will talk about the geometric distribution, extract its formula, discuss its mean, and give examples.

## Explanation of Geometric Distribution

The geometric distribution is a discrete probability distribution that narrates the amount of trials needed to reach the first success in a sequence of Bernoulli trials. A Bernoulli trial is a trial which has two viable results, usually referred to as success and failure. For example, flipping a coin is a Bernoulli trial because it can either turn out to be heads (success) or tails (failure).

The geometric distribution is utilized when the experiments are independent, which means that the outcome of one trial doesn’t affect the result of the upcoming test. Additionally, the chances of success remains constant across all the tests. 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 trials needed to achieve the first success, k is the count of experiments needed to obtain 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 defined as the anticipated value of the amount of trials required to achieve 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 likely count of tests required to get the first success. For instance, if the probability of success is 0.5, therefore we anticipate to obtain the first success after two trials on average.

## Examples of Geometric Distribution

Here are some essential examples of geometric distribution

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

Suppose we flip an honest coin until the first head turns up. 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 that portrays the count of coin flips required to achieve the first 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 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.

Let’s assume we roll a fair die up until the initial six shows up. The probability of success (obtaining a six) is 1/6, and the probability of failure (getting all other number) is 5/6. Let X be the irregular variable which represents the number of die rolls required to get 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 obtaining 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 achieving 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 achieving 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 broad range of practical phenomena, for example the number of tests needed to obtain the first success in different scenarios.

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