# Geometric Distribution - Definition, Formula, Mean, Examples

Probability theory is a important branch of mathematics which deals with the study of random occurrence. One of the important theories in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution that models the number of trials needed to obtain the initial success in a secession of Bernoulli trials. In this blog, we will define the geometric distribution, extract its formula, discuss its mean, and provide examples.

## Explanation of Geometric Distribution

The geometric distribution is a discrete probability distribution which narrates the amount of experiments required to achieve the first success in a series of Bernoulli trials. A Bernoulli trial is a test which has two viable results, typically referred 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 trials are independent, which means that the outcome of one experiment doesn’t affect the outcome of the upcoming test. Furthermore, the probability of success remains unchanged across all the trials. We could signify 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 depicts the number of trials required to achieve the initial success, k is the number 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 explained as the expected value of the amount of test required to get 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 number of trials needed to obtain the first success. For instance, if the probability of success is 0.5, then we expect to get the initial success after two trials on average.

## Examples of Geometric Distribution

Here are few primary examples of geometric distribution

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

Imagine we flip a fair coin till the initial head turns up. The probability of success (getting a head) is 0.5, and the probability of failure (getting a tail) is as well as 0.5. Let X be the random variable which depicts the count of coin flips needed 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 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 getting 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 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 a fair die up until the initial six shows up.

Let’s assume we roll an honest die till the first six turns up. The probability of success (getting a six) is 1/6, and the probability of failure (getting all other number) is 5/6. Let X be the random variable which depicts the count of die rolls required to achieve the first 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 achieving 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 getting 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 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 important concept in probability theory. It is applied to model a broad array of practical phenomena, for example the count of experiments needed to obtain the initial success in various scenarios.

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