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 that models the number of tests needed to get the first success in a sequence of Bernoulli trials. In this blog, 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 portrays the amount of trials required to reach the first success in a succession of Bernoulli trials. A Bernoulli trial is a test which has two possible outcomes, generally indicated to as success and failure. For instance, 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 trials are independent, meaning that the result of one test doesn’t affect the result of the upcoming test. Furthermore, the chances of success remains same throughout all the trials. 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 provided by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable that represents the number of trials required to attain the first success, k is the count of experiments needed 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 described as the anticipated value of the amount of test needed to get 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 likely count of experiments needed to achieve the initial success. For example, if the probability of success is 0.5, therefore we anticipate to obtain the initial success following two trials on average.
Examples of Geometric Distribution
Here are handful of primary examples of geometric distribution
Example 1: Flipping a fair coin till the first head turn up.
Let’s assume we toss an honest coin till the initial head appears. 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 portrays the number of coin flips needed to get 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 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 getting the first 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 turns up.
Let’s assume we roll a fair die till the initial six shows up. The probability of success (achieving a six) is 1/6, and the probability of failure (achieving any other number) is 5/6. Let X be the random variable which represents the number of die rolls needed to achieve 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 achieving 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 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 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 crucial concept in probability theory. It is utilized to model a broad array of real-life scenario, for instance the number of tests required to obtain the initial success in different situations.
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