# Geometric Distribution - Definition, Formula, Mean, Examples

Probability theory is an essential division of mathematics 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 discrete probability distribution that models the amount of tests needed to get the first success in a series of Bernoulli trials. In this blog, we will define the geometric distribution, derive its formula, discuss its mean, and offer examples.

## Definition of Geometric Distribution

The geometric distribution is a discrete probability distribution which narrates the number of trials needed to reach the first success in a sequence of Bernoulli trials. A Bernoulli trial is a trial that has two possible outcomes, generally referred to as success and failure. Such as tossing a coin is a Bernoulli trial because it can likewise turn out to be heads (success) or tails (failure).

The geometric distribution is used when the trials are independent, meaning that the consequence of one trial doesn’t impact the result of the next trial. In addition, the chances 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 specified by the formula:

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

Where X is the random variable that portrays the number of test needed to get the initial success, k is the count of trials needed to obtain 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 described as the anticipated value of the number of test needed to get the first success. The mean is stated in 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 likely count of trials required to achieve the initial success. Such as if the probability of success is 0.5, then we expect to obtain the initial success following two trials on average.

## Examples of Geometric Distribution

Here are handful of essential examples of geometric distribution

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

Imagine we toss an honest coin till the initial head shows up. The probability of success (obtaining 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 portrays 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 achieving 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 getting 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 on.

Example 2: Rolling an honest die until the first six appears.

Suppose we roll a fair die up until the first six appears. The probability of success (achieving a six) is 1/6, and the probability of failure (obtaining all other number) is 5/6. Let X be the irregular variable that represents the count of die rolls needed to obtain 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 first 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 obtaining the initial 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 used to model a wide range of real-world scenario, such as the number of experiments needed to obtain the initial success in several situations.

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