Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is an essential division of mathematics that deals with the study of random occurrence. One of the crucial theories in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution which models the number of experiments needed to get the first success in a sequence of Bernoulli trials. In this article, we will explain the geometric distribution, extract 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 reach the initial success in a succession of Bernoulli trials. A Bernoulli trial is a trial that has two viable outcomes, typically indicated to as success and failure. Such as tossing a coin is a Bernoulli trial because it can likewise turn out to be heads (success) or tails (failure).
The geometric distribution is utilized when the trials are independent, which means that the outcome of one test does not affect the outcome of the next trial. Additionally, the probability of success remains constant across 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 portrays the amount of test required to achieve the first success, k is the count of trials required to obtain 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 number of test required 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 number of trials needed to get the initial success. Such as if the probability of success is 0.5, then we expect to get 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 shows up.
Let’s assume we toss an honest coin until the initial head shows up. The probability of success (getting 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 that portrays the number of coin flips needed to obtain 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 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 forth.
Example 2: Rolling an honest die until the first six shows up.
Suppose we roll a fair die till the first six turns up. The probability of success (getting a six) is 1/6, and the probability of failure (achieving any other number) is 5/6. Let X be the random variable which depicts the number of die rolls needed to obtain the initial 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 getting 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 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 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 on.
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The geometric distribution is a important concept in probability theory. It is utilized to model a broad array of real-life scenario, such as the count of tests required to obtain the initial success in several situations.
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