Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is a important division of math which deals with the study of random occurrence. One of the crucial ideas in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution that models the amount of tests needed to get the first success in a series of Bernoulli trials. In this blog article, we will talk about the geometric distribution, derive its formula, discuss its mean, and provide examples.
Definition of Geometric Distribution
The geometric distribution is a discrete probability distribution which narrates the number of trials required to achieve the initial success in a series of Bernoulli trials. A Bernoulli trial is an experiment that has two likely results, usually referred to as success and failure. For example, tossing a coin is a Bernoulli trial because it can likewise come up heads (success) or tails (failure).
The geometric distribution is used when the experiments are independent, meaning that the result of one trial doesn’t impact the outcome of the upcoming test. Furthermore, the probability of success remains unchanged throughout all the tests. We could 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 that depicts the amount of test needed to get the first 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 described as the expected value of the number of experiments 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 expected number of tests required to get the initial success. Such as if the probability of success is 0.5, therefore we expect to attain the first success after two trials on average.
Examples of Geometric Distribution
Here are few essential examples of geometric distribution
Example 1: Flipping a fair coin till the first head appears.
Suppose we flip 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 also 0.5. Let X be the random variable that represents the number of coin flips required to achieve the initial 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 first 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 until the first six shows up.
Suppose we roll an honest die up until the first six appears. 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 irregular variable that depicts the count of die rolls needed to obtain 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 getting the first six on the initial roll is:
P(X = 1) = (5/6)^(1-1) * 1/6 = 1/6
For k = 2, the probability of achieving 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 obtaining the first 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 a important concept in probability theory. It is used to model a broad array of real-world phenomena, such as the number of experiments required to get the initial success in different situations.
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