Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is an essential division of math that handles the study of random events. One of the essential theories in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution which models the amount of trials needed to get the first success in a series of Bernoulli trials. In this blog, we will talk about the geometric distribution, derive its formula, discuss its mean, and give examples.
Definition of Geometric Distribution
The geometric distribution is a discrete probability distribution that portrays the number of experiments needed to accomplish the initial success in a series of Bernoulli trials. A Bernoulli trial is an experiment which has two viable outcomes, generally referred to as success and failure. For instance, 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 outcome of one trial doesn’t impact the result of the upcoming trial. Additionally, the probability of success remains same throughout all the tests. We could 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 specified by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable that represents the number of test required to attain the initial success, k is the count of trials required to attain the initial 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 explained as the expected value of the amount of trials needed to get 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 expected count of tests required to get the first success. For example, if the probability of success is 0.5, therefore we anticipate to get the initial success after two trials on average.
Examples of Geometric Distribution
Here are some basic examples of geometric distribution
Example 1: Flipping a fair coin till the first head turn up.
Imagine we toss a fair coin until the first head turns up. The probability of success (getting a head) is 0.5, and the probability of failure (obtaining a tail) is also 0.5. Let X be the random variable which 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 obtaining 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 achieving the initial 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 initial six shows up.
Let’s assume we roll an honest die until the initial six shows up. The probability of success (obtaining a six) is 1/6, and the probability of failure (obtaining any other number) is 5/6. Let X be the irregular variable that portrays the number of die rolls required 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 first 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 obtaining 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 crucial theory in probability theory. It is used to model a wide array of real-life phenomena, for instance the number of trials needed to achieve the first success in different scenarios.
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