# Geometric Distribution - Definition, Formula, Mean, Examples

Probability theory is an essential division of math which handles the study of random occurrence. One of the important ideas in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution that models the amount of experiments required to obtain the initial success in a secession of Bernoulli trials. In this blog article, we will explain the geometric distribution, extract its formula, discuss its mean, and give examples.

## Definition of Geometric Distribution

The geometric distribution is a discrete probability distribution that narrates the amount of experiments required to reach the initial success in a series of Bernoulli trials. A Bernoulli trial is a test which has two viable outcomes, typically referred to as success and failure. For example, flipping a coin is a Bernoulli trial because it can either come up heads (success) or tails (failure).

The geometric distribution is utilized when the trials are independent, meaning that the consequence of one trial does not impact the outcome of the upcoming test. In addition, the chances of success remains unchanged throughout all the tests. 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 that represents the number of trials required to attain the initial success, k is the number of experiments needed to attain 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 defined as the expected value of the amount of trials needed to achieve 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 get the initial success. For instance, 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 some essential examples of geometric distribution

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

Suppose we toss a fair coin till the first head shows 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 which represents the count of coin flips needed 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 first 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 achieving 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 a fair die up until the first six appears.

Suppose we roll a fair die up until the initial six turns up. 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 that depicts the number of die rolls needed to get the initial six. The PMF of X is given by:

P(X = k) = (1 - 1/6)^(k-1) * 1/6 = (5/6)^(k-1) * 1/6

For k = 1, the probability of getting the initial 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 achieving 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 range of practical phenomena, for example the number of experiments required to obtain the initial success in different situations.

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