# Geometric Distribution - Definition, Formula, Mean, Examples

Probability theory is ac crucial division of math which deals with the study of random occurrence. One of the essential theories in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution that models the amount of tests required to get the initial success in a secession of Bernoulli trials. In this blog article, we will define the geometric distribution, extract its formula, discuss its mean, and provide examples.

## Meaning of Geometric Distribution

The geometric distribution is a discrete probability distribution which portrays the number of experiments needed to accomplish the initial success in a sequence of Bernoulli trials. A Bernoulli trial is an experiment which has two likely results, generally referred to as success and failure. For instance, tossing a coin is a Bernoulli trial since it can either come up heads (success) or tails (failure).

The geometric distribution is applied when the experiments are independent, meaning that the result of one experiment doesn’t impact the result of the upcoming test. Furthermore, the chances of success remains same throughout all the trials. 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 which portrays the amount of trials required to attain the first success, k is the count of tests needed to attain the first 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 defined as the anticipated value of the amount 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 a single Bernoulli trial.

The mean is the likely number of tests required to achieve the first success. For example, if the probability of success is 0.5, therefore we expect to attain the first success following two trials on average.

## Examples of Geometric Distribution

Here are few basic examples of geometric distribution

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

Let’s assume we toss an honest coin till the first head turns up. The probability of success (getting a head) is 0.5, and the probability of failure (getting a tail) is also 0.5. Let X be the random variable that portrays the number of coin flips required to achieve the first head. The PMF of X is stated 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 achieving 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 getting 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 until the initial six appears.

Let’s assume we roll a fair die up until the initial six appears. The probability of success (obtaining a six) is 1/6, and the probability of failure (achieving all other number) is 5/6. Let X be the random variable which represents the number 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 initial 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 on.

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The geometric distribution is a important concept in probability theory. It is utilized to model a wide range of real-world phenomena, such as the count of experiments required to achieve the initial success in several scenarios.

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