Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is an essential division of math which takes up the study of random occurrence. One of the crucial concepts in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution that models the amount of experiments required to get the first success in a secession of Bernoulli trials. In this article, we will talk about 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 number of tests required to achieve the first success in a series of Bernoulli trials. A Bernoulli trial is a trial which has two likely results, typically indicated to as success and failure. For example, flipping a coin is a Bernoulli trial since it can likewise turn out to be heads (success) or tails (failure).
The geometric distribution is used when the experiments are independent, which means that the result of one trial does not affect the outcome of the upcoming test. Furthermore, the chances of success remains unchanged throughout all the trials. 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 given by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable that represents the amount of trials needed to attain the initial success, k is the count of trials required to achieve the initial 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 likely value of the amount of experiments required 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 an individual Bernoulli trial.
The mean is the anticipated count of tests required to achieve the initial success. Such as if the probability of success is 0.5, therefore we anticipate to attain the initial success following two trials on average.
Examples of Geometric Distribution
Here are handful of basic examples of geometric distribution
Example 1: Flipping a fair coin up until the first head shows up.
Imagine we flip 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 which portrays the count of coin flips required to achieve 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 first 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 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 forth.
Example 2: Rolling an honest die until the first six appears.
Suppose we roll an honest die until the initial six shows up. 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 number of die rolls required to achieve the first six. The PMF of X is given by:
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 obtaining 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 an essential theory in probability theory. It is applied to model a wide range of real-life phenomena, such as the number of trials required to obtain the initial success in various situations.
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