# Geometric Distribution - Definition, Formula, Mean, Examples

Probability theory is a important branch of math which handles the study of random events. One of the important ideas in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution which models the amount of experiments needed to obtain the first success in a series of Bernoulli trials. In this blog article, we will explain the geometric distribution, derive its formula, discuss its mean, and offer examples.

## Meaning of Geometric Distribution

The geometric distribution is a discrete probability distribution that describes the amount of trials needed to reach the first 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, flipping a coin is a Bernoulli trial since it can either come up heads (success) or tails (failure).

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

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

Where X is the random variable which portrays the number of test required to attain the initial success, k is the number of tests required to obtain the first 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 anticipated value of the number of test needed to obtain 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 anticipated count of experiments required to get the first success. For example, if the probability of success is 0.5, then we anticipate to obtain the first success after two trials on average.

## Examples of Geometric Distribution

Here are some essential examples of geometric distribution

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

Let’s assume we toss a fair 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 as well as 0.5. Let X be the random variable which depicts 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 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 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 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 initial six appears.

Let’s assume we roll an honest die till the initial six shows up. 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 random variable that portrays the number of die rolls needed to achieve 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 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 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 crucial concept in probability theory. It is applied to model a broad range of real-world scenario, such as the count of tests required to achieve the first success in different scenarios.

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