# Geometric Distribution - Definition, Formula, Mean, Examples

Probability theory is a important division of mathematics which handles the study of random events. One of the important ideas in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution that models the number of tests needed to obtain the initial success in a series of Bernoulli trials. In this article, we will define the geometric distribution, derive its formula, discuss its mean, and offer examples.

## Explanation of Geometric Distribution

The geometric distribution is a discrete probability distribution that narrates the number of tests required to reach the first success in a series of Bernoulli trials. A Bernoulli trial is an experiment that has two likely outcomes, typically referred to as success and failure. For instance, tossing a coin is a Bernoulli trial since it can likewise come up heads (success) or tails (failure).

The geometric distribution is utilized when the trials are independent, meaning that the outcome of one test does not affect the outcome of the upcoming test. In addition, the probability of success remains same throughout all the tests. We can 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 that portrays the amount of trials required to achieve the initial success, k is the number of experiments needed 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 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 an individual Bernoulli trial.

The mean is the expected count of tests required to obtain the initial success. For instance, 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 handful of basic examples of geometric distribution

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

Suppose we flip a fair coin until the initial head appears. The probability of success (getting a head) is 0.5, and the probability of failure (obtaining a tail) is as well as 0.5. Let X be the random variable that represents the count of coin flips required to achieve 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 achieving 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 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 until the initial six appears.

Suppose we roll a fair die up until the initial six turns 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 irregular variable that represents the number of die rolls required to achieve the initial six. The PMF of X is provided 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 initial six on the initial 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 achieving 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 utilized to model a broad range of practical scenario, for example the number of tests required to get the first success in different situations.

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