# Geometric Distribution - Definition, Formula, Mean, Examples

Probability theory is a important department of math which handles the study of random events. One of the crucial ideas in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution that models the number of experiments required to obtain the initial success in a sequence of Bernoulli trials. In this blog article, we will define the geometric distribution, derive its formula, discuss its mean, and offer examples.

## Meaning of Geometric Distribution

The geometric distribution is a discrete probability distribution which narrates the number of trials needed to achieve the initial success in a series of Bernoulli trials. A Bernoulli trial is a test which has two likely outcomes, typically indicated to as success and failure. For example, tossing a coin is a Bernoulli trial because it can either come up heads (success) or tails (failure).

The geometric distribution is used when the tests are independent, meaning that the consequence of one test does not impact the outcome of the next test. Additionally, the probability of success remains constant across 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 which represents the number of test required to get the initial success, k is the count of tests required 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 explained as the anticipated value of the number of experiments needed to get the initial success. The mean is stated in 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 anticipated number of tests needed to get the initial success. For example, if the probability of success is 0.5, therefore we expect to obtain 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 until the first head shows up.

Suppose we flip a fair coin till the first head turns up. 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 depicts the number of coin flips required to get the initial 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 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 achieving the first head on the third flip is:

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

And so forth.

Example 2: Rolling an honest 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 (achieving a six) is 1/6, and the probability of failure (achieving any other number) is 5/6. Let X be the random variable that represents the number of die rolls needed to achieve 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 achieving 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 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 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 concept in probability theory. It is applied to model a broad array of real-life phenomena, for example the count of experiments needed to achieve the initial success in several situations.

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