Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is an essential department of math which takes up the study of random occurrence. One of the important theories in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution which models the number of trials needed to get the first success in a secession of Bernoulli trials. In this blog, we will define the geometric distribution, derive its formula, discuss its mean, and offer examples.
Meaning of Geometric Distribution
The geometric distribution is a discrete probability distribution which narrates the amount of tests required to achieve the initial success in a series of Bernoulli trials. A Bernoulli trial is a test that has two viable outcomes, generally indicated to as success and failure. For example, flipping a coin is a Bernoulli trial because it can likewise come up heads (success) or tails (failure).
The geometric distribution is used when the tests are independent, meaning that the result of one trial doesn’t impact the outcome of the next trial. Additionally, the chances 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 which represents the amount of test needed to achieve the first success, k is the number of trials required to attain 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 explained as the expected value of the number of test needed to obtain 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 an individual Bernoulli trial.
The mean is the likely count of trials needed to obtain the initial success. For example, if the probability of success is 0.5, therefore we expect to attain the first success following two trials on average.
Examples of Geometric Distribution
Here are handful of primary examples of geometric distribution
Example 1: Tossing a fair coin till the first head appears.
Let’s assume we toss an honest coin until the initial head shows up. The probability of success (obtaining 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 count of coin flips required to obtain 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 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 getting the first 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 till the initial six appears.
Suppose we roll an honest die until the initial six turns up. 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 portrays the number of die rolls required to obtain the first 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 obtaining 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 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 achieving the initial 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 an essential theory in probability theory. It is applied to model a wide range of real-world phenomena, for instance the count of experiments needed to obtain the initial success in several situations.
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