Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is a important branch of math which takes up the study of random events. One of the crucial ideas in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution that models the number of experiments needed to get 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 provide examples.
Meaning of Geometric Distribution
The geometric distribution is a discrete probability distribution which narrates the amount of experiments needed to achieve the initial success in a succession of Bernoulli trials. A Bernoulli trial is an experiment that has two possible results, usually indicated to as success and failure. Such as flipping a coin is a Bernoulli trial since it can either come up heads (success) or tails (failure).
The geometric distribution is used when the trials are independent, which means that the outcome of one test does not impact the outcome of the upcoming test. Additionally, the chances of success remains constant across all the trials. We can indicate 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 portrays the amount of test needed to get the initial success, k is the count of trials required to obtain the initial 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 explained as the likely value of the amount of experiments required 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 a single Bernoulli trial.
The mean is the likely number of tests needed to get the initial success. For example, if the probability of success is 0.5, then we expect to attain the first success after two trials on average.
Examples of Geometric Distribution
Here are few basic examples of geometric distribution
Example 1: Flipping a fair coin until the first head appears.
Imagine we flip a fair coin till 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 represents the number of coin flips required to get the first 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 getting 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 up until the first six turns up.
Let’s assume we roll a fair die until the initial six turns up. The probability of success (obtaining a six) is 1/6, and the probability of failure (getting all other number) is 5/6. Let X be the irregular variable that represents the count of die rolls required to get the first 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 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 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 an essential concept in probability theory. It is applied to model a broad array of real-world phenomena, such as the count of tests required to get the initial success in different situations.
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