# Geometric Distribution - Definition, Formula, Mean, Examples

Probability theory is ac crucial division of mathematics which takes up the study of random events. One of the essential ideas in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution which models the number of trials needed to get the first success in a secession of Bernoulli trials. In this 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 describes the number of experiments needed to reach the first success in a series of Bernoulli trials. A Bernoulli trial is a trial which has two viable outcomes, generally indicated to as success and failure. For instance, flipping a coin is a Bernoulli trial since it can either come up heads (success) or tails (failure).

The geometric distribution is used when the tests are independent, which means that the result of one test doesn’t impact the outcome of the upcoming trial. Furthermore, the probability of success remains same throughout all the tests. We could 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 provided by the formula:

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

Where X is the random variable which depicts the number of test required to achieve the initial success, k is the count of tests 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 experiments required to achieve 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 a single Bernoulli trial.

The mean is the likely count of tests required to obtain the first success. For instance, if the probability of success is 0.5, therefore we anticipate to attain the initial success after two trials on average.

## Examples of Geometric Distribution

Here are few basic examples of geometric distribution

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

Imagine we toss a fair coin till the first 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 depicts the count of coin flips required to achieve 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 obtaining 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 up until the first six turns up. The probability of success (getting a six) is 1/6, and the probability of failure (achieving all other number) is 5/6. Let X be the irregular variable which depicts the count 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 initial 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 on.

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The geometric distribution is a important concept in probability theory. It is utilized to model a wide array of real-world scenario, for example the count of tests required to achieve the initial success in various situations.

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