Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is ac crucial department of math which deals with the study of random events. One of the important theories in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution which models the amount of tests required to get the first success in a sequence of Bernoulli trials. In this article, we will talk about the geometric distribution, extract its formula, discuss its mean, and give examples.
Explanation of Geometric Distribution
The geometric distribution is a discrete probability distribution that narrates the amount of experiments needed to achieve the initial success in a succession of Bernoulli trials. A Bernoulli trial is a test which has two possible results, usually 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 applied when the tests are independent, meaning that the consequence of one trial does not impact the result of the next trial. Additionally, the probability 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 number of test required to attain the initial success, k is the count of experiments required to attain 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 expected value of the amount of test needed to get 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 initial success. For instance, if the probability of success is 0.5, then we expect to obtain the initial success after two trials on average.
Examples of Geometric Distribution
Here are few basic examples of geometric distribution
Example 1: Flipping a fair coin till the first head turn up.
Let’s assume we flip a fair coin until the first head turns up. The probability of success (obtaining a head) is 0.5, and the probability of failure (getting a tail) is also 0.5. Let X be the random variable that represents the number of coin flips required to achieve the first head. The PMF of X is given by:
P(X = k) = (1 - 0.5)^(k-1) * 0.5 = 0.5^(k-1) * 0.5
For k = 1, the probability of getting 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 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 getting 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 till the initial six shows up.
Let’s assume we roll an honest die up until the first six appears. The probability of success (achieving a six) is 1/6, and the probability of failure (obtaining any other number) is 5/6. Let X be the random variable which depicts the number of die rolls needed to achieve 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 achieving the first six on the first roll is:
P(X = 1) = (5/6)^(1-1) * 1/6 = 1/6
For k = 2, the probability of obtaining 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 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 an essential concept in probability theory. It is utilized to model a wide range of real-world phenomena, for instance the count of tests needed to obtain the initial success in different situations.
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