Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is an essential department of math that deals with the study of random events. One of the essential concepts in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution which models the amount of experiments required to obtain the first success in a series of Bernoulli trials. In this blog, we will explain the geometric distribution, derive its formula, discuss its mean, and provide examples.
Definition of Geometric Distribution
The geometric distribution is a discrete probability distribution that describes the number of experiments needed to achieve the initial success in a series of Bernoulli trials. A Bernoulli trial is a trial that has two likely results, typically referred to as success and failure. For example, flipping a coin is a Bernoulli trial since it can either come up heads (success) or tails (failure).
The geometric distribution is applied when the experiments are independent, which means that the outcome of one test does not impact the result of the upcoming test. Furthermore, the probability of success remains constant throughout 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 which represents the number of test required to achieve the initial success, k is the count of tests needed 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 defined as the anticipated value of the number of experiments 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 an individual Bernoulli trial.
The mean is the expected count of trials required to get the first success. For instance, if the probability of success is 0.5, then we expect to get the first success following two trials on average.
Examples of Geometric Distribution
Here are handful of primary examples of geometric distribution
Example 1: Flipping a fair coin until the first head shows up.
Let’s assume we toss an honest coin till the initial head shows up. The probability of success (getting a head) is 0.5, and the probability of failure (getting a tail) is also 0.5. Let X be the random variable which represents the number of coin flips needed to achieve the initial 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 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 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 on.
Example 2: Rolling a fair die up until the first six shows up.
Suppose we roll an honest die till the initial six shows up. The probability of success (obtaining a six) is 1/6, and the probability of failure (achieving any other number) is 5/6. Let X be the random variable that depicts the number of die rolls required to get the first six. The PMF of X is stated as:
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 getting 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 first 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 an essential concept in probability theory. It is applied to model a wide range of practical phenomena, such as the count of experiments required to obtain the initial success in various scenarios.
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