Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is an essential division of mathematics which deals with the study of random events. 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 obtain 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 offer examples.
Meaning of Geometric Distribution
The geometric distribution is a discrete probability distribution that describes the amount of experiments required to achieve the first success in a succession of Bernoulli trials. A Bernoulli trial is a trial which has two possible outcomes, usually indicated to as success and failure. Such as flipping a coin is a Bernoulli trial since it can either turn out to be heads (success) or tails (failure).
The geometric distribution is used when the experiments are independent, which means that the consequence of one test doesn’t impact the result of the next trial. Additionally, the chances of success remains unchanged 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 represents the amount of test required to get the first success, k is the number of trials needed to obtain 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 expected value of the number of trials needed to achieve the first 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 number of experiments required to get the first success. For example, if the probability of success is 0.5, then we anticipate to obtain the initial 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 until the first head appears.
Imagine we toss an honest coin till the first head turns 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 that depicts the count 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 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 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 till the first six shows up.
Let’s assume we roll an honest die up until the first six appears. The probability of success (achieving a six) is 1/6, and the probability of failure (obtaining any other number) is 5/6. Let X be the irregular variable that depicts the count of die rolls required to get the initial six. The PMF of X is provided 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 first 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 theory in probability theory. It is applied to model a wide range of practical phenomena, such as the number of experiments required to get the initial success in several situations.
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