Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is a important division of mathematics which takes up the study of random occurrence. One of the important theories in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution that models the amount of tests required to get the initial success in a secession of Bernoulli trials. In this blog, we will explain the geometric distribution, extract its formula, discuss its mean, and provide examples.
Definition of Geometric Distribution
The geometric distribution is a discrete probability distribution that describes the amount of experiments needed to reach the initial success in a series of Bernoulli trials. A Bernoulli trial is an experiment which has two possible outcomes, typically indicated to as success and failure. Such as flipping a coin is a Bernoulli trial because it can either turn out to be heads (success) or tails (failure).
The geometric distribution is utilized when the experiments are independent, which means that the outcome of one test does not affect the result of the upcoming test. Furthermore, the chances of success remains constant throughout all the trials. We can 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 which depicts the amount of trials required to achieve the initial success, k is the count of tests needed to achieve 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 defined as the likely value of the amount of trials needed to obtain the first 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 likely count of tests required to obtain the first success. Such as if the probability of success is 0.5, then we anticipate to get the initial success after two trials on average.
Examples of Geometric Distribution
Here are few essential examples of geometric distribution
Example 1: Tossing a fair coin up until the first head turn up.
Let’s assume we toss an honest coin till the initial head turns up. The probability of success (getting a head) is 0.5, and the probability of failure (obtaining a tail) is as well as 0.5. Let X be the random variable which portrays the number of coin flips required to achieve 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 obtaining 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 getting 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 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 till the initial six appears.
Suppose we roll a fair die till the first six appears. The probability of success (obtaining a six) is 1/6, and the probability of failure (obtaining all other number) is 5/6. Let X be the irregular variable which represents 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 obtaining 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 obtaining 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 achieving 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 theory in probability theory. It is used to model a wide array of real-life scenario, such as the count of tests needed to get the first success in different scenarios.
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