Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is ac crucial division of math which deals with the study of random occurrence. One of the crucial theories in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution that models the number of tests needed to get the first success in a sequence of Bernoulli trials. In this blog, we will explain the geometric distribution, derive its formula, discuss its mean, and provide examples.
Explanation of Geometric Distribution
The geometric distribution is a discrete probability distribution that describes the number of tests required to achieve the initial success in a succession of Bernoulli trials. A Bernoulli trial is a test which has two possible results, generally indicated to as success and failure. For instance, flipping a coin is a Bernoulli trial since it can likewise come up heads (success) or tails (failure).
The geometric distribution is applied when the tests are independent, which means that the result of one test does not affect the result of the upcoming trial. Additionally, the chances of success remains same throughout all the tests. 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 given by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable that depicts the number of test required to get the first success, k is the number of experiments needed to obtain 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 described as the likely value of the number of trials needed to obtain the initial 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 needed to obtain the first success. Such as if the probability of success is 0.5, then we expect to attain the first success following two trials on average.
Examples of Geometric Distribution
Here are handful of essential examples of geometric distribution
Example 1: Flipping a fair coin till the first head shows up.
Let’s assume we toss a fair coin until the first head appears. The probability of success (obtaining 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 which represents the count of coin flips required to achieve the first head. The PMF of X is stated as:
P(X = k) = (1 - 0.5)^(k-1) * 0.5 = 0.5^(k-1) * 0.5
For k = 1, the probability of getting 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 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 until the initial six appears.
Suppose we roll a fair die till the first six appears. The probability of success (achieving a six) is 1/6, and the probability of failure (achieving any other number) is 5/6. Let X be the irregular variable which 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 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 an essential concept in probability theory. It is used to model a broad range of practical phenomena, for instance the number of tests needed to get the first success in different situations.
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