# Geometric Distribution - Definition, Formula, Mean, Examples

Probability theory is ac crucial department of mathematics that takes up 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 which models the amount of experiments needed to obtain the first success in a secession of Bernoulli trials. In this blog article, we will explain the geometric distribution, extract its formula, discuss its mean, and offer examples.

## Meaning of Geometric Distribution

The geometric distribution is a discrete probability distribution which portrays the amount of trials needed to accomplish the first success in a sequence of Bernoulli trials. A Bernoulli trial is an experiment which has two viable outcomes, generally indicated to as success and failure. For instance, tossing a coin is a Bernoulli trial since it can likewise come up heads (success) or tails (failure).

The geometric distribution is utilized when the tests are independent, meaning that the result of one test doesn’t affect the outcome of the next test. Additionally, the probability of success remains same across all the tests. 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 specified by the formula:

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

Where X is the random variable that portrays the amount of test needed to get the initial success, k is the number of tests required to attain 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 trials needed to obtain the initial 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 anticipated count of experiments required to achieve the initial success. For example, if the probability of success is 0.5, therefore we anticipate to attain 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 till the first head appears.

Suppose we flip an honest coin till the initial head shows up. The probability of success (obtaining a head) is 0.5, and the probability of failure (obtaining a tail) is also 0.5. Let X be the random variable that depicts the count 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 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 on.

Example 2: Rolling a fair die up until the first six turns up.

Let’s assume we roll a fair die until the first six appears. The probability of success (getting a six) is 1/6, and the probability of failure (obtaining all other number) is 5/6. Let X be the irregular 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 getting 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 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 forth.

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The geometric distribution is a crucial theory in probability theory. It is utilized to model a wide array of practical scenario, for example the number of experiments needed to obtain the first success in several situations.

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