# Geometric Distribution - Definition, Formula, Mean, Examples

Probability theory is an essential department of math which deals with 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 required to obtain the initial success in a sequence of Bernoulli trials. In this blog, we will define the geometric distribution, derive its formula, discuss its mean, and provide examples.

## Explanation of Geometric Distribution

The geometric distribution is a discrete probability distribution which portrays the amount of trials needed to achieve the initial success in a sequence of Bernoulli trials. A Bernoulli trial is a test which has two likely outcomes, usually referred to as success and failure. Such as tossing a coin is a Bernoulli trial since 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 outcome of one trial doesn’t affect the result of the next test. Additionally, the chances of success remains constant across all the trials. We can 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 portrays the amount of trials required to achieve the initial 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 described as the expected value of the number of trials needed to obtain 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 anticipated number of tests required to obtain the first success. For instance, if the probability of success is 0.5, then we expect to get the first success after two trials on average.

## Examples of Geometric Distribution

Here are handful of primary examples of geometric distribution

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

Suppose we flip an honest coin until the first head appears. 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 that portrays the number of coin flips required to obtain 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 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 obtaining 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 until the first six turns up.

Let’s assume we roll a fair die up until the initial six shows up. The probability of success (obtaining a six) is 1/6, and the probability of failure (achieving any other number) is 5/6. Let X be the random variable which depicts the number of die rolls required 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 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 getting 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 applied to model a broad array of real-world scenario, for example the number of trials needed to get the first success in various situations.

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