# Geometric Distribution - Definition, Formula, Mean, Examples

Probability theory is ac crucial branch of math which deals with the study of random occurrence. One of the important concepts in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution which models the amount of tests required to get the first success in a sequence of Bernoulli trials. In this blog article, we will define the geometric distribution, derive its formula, discuss its mean, and give examples.

## Explanation of Geometric Distribution

The geometric distribution is a discrete probability distribution that portrays the number of tests required to achieve the first success in a series of Bernoulli trials. A Bernoulli trial is a trial which has two possible results, generally indicated to as success and failure. Such as flipping a coin is a Bernoulli trial because it can likewise come up heads (success) or tails (failure).

The geometric distribution is utilized when the tests are independent, meaning that the outcome of one experiment doesn’t impact the outcome of the next trial. Furthermore, the probability of success remains unchanged throughout 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 specified by the formula:

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

Where X is the random variable that depicts the number of trials required to attain the first success, k is the count of tests needed to achieve 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 a single Bernoulli trial.

The mean is the likely number of trials needed to achieve the first success. For example, if the probability of success is 0.5, therefore we expect to get the initial success after two trials on average.

## Examples of Geometric Distribution

Here are some primary examples of geometric distribution

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

Let’s assume 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 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 first head on the third flip is:

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

And so on.

Example 2: Rolling an honest die up until the initial six turns up.

Suppose we roll an honest die 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 irregular variable which depicts the number of die rolls needed to obtain 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 getting 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 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 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 a important theory in probability theory. It is used to model a broad array of real-world scenario, such as the number of trials required to achieve the initial success in various situations.

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