Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is a important branch of math which takes up 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 that models the number of tests required to get the first success in a sequence of Bernoulli trials. In this blog article, we will define the geometric distribution, derive its formula, discuss its mean, and provide examples.
Meaning of Geometric Distribution
The geometric distribution is a discrete probability distribution which narrates the amount of tests required to accomplish the initial success in a succession of Bernoulli trials. A Bernoulli trial is a trial that has two likely results, usually referred to as success and failure. Such as tossing a coin is a Bernoulli trial because it can likewise come up heads (success) or tails (failure).
The geometric distribution is utilized when the experiments are independent, meaning that the result of one experiment doesn’t impact the outcome of the upcoming trial. Furthermore, the probability of success remains same across all the tests. We can 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 specified by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable which depicts the amount of test required to achieve the first success, k is the count of tests required 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 explained as the expected value of the number of experiments required to obtain the initial 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 anticipated count of experiments needed to get the initial success. For instance, if the probability of success is 0.5, then we anticipate to attain the initial success after two trials on average.
Examples of Geometric Distribution
Here are few essential examples of geometric distribution
Example 1: Tossing a fair coin until the first head turn up.
Imagine we flip a fair coin till the first 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 depicts the number of coin flips needed to get the first head. The PMF of X is given by:
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 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 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 an honest die till the first six turns up.
Suppose we roll a fair die until the first six appears. The probability of success (obtaining a six) is 1/6, and the probability of failure (getting all other number) is 5/6. Let X be the irregular variable that depicts the number of die rolls needed to achieve 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 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 obtaining 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 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 a crucial theory in probability theory. It is applied to model a wide range of real-world phenomena, for instance the number of experiments needed to achieve the first success in several situations.
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