Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is ac crucial division of mathematics which deals with the study of random events. One of the essential ideas in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution that models the amount of tests required 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 give examples.
Meaning of Geometric Distribution
The geometric distribution is a discrete probability distribution which portrays the number of experiments required to reach the first success in a series of Bernoulli trials. A Bernoulli trial is a test that has two possible outcomes, usually indicated to as success and failure. For instance, flipping a coin is a Bernoulli trial because it can likewise turn out to be heads (success) or tails (failure).
The geometric distribution is utilized when the tests are independent, meaning that the outcome of one experiment does not impact the outcome of the upcoming test. Furthermore, the probability of success remains same throughout all the tests. We could 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 specified by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable that depicts the number of trials required to get the first success, k is the number of trials required to obtain 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 defined as the anticipated value of the number of trials required to achieve 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 tests required to achieve the first success. Such as if the probability of success is 0.5, therefore we expect to obtain 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 till the first head appears.
Imagine we flip a fair coin till the initial head turns up. The probability of success (getting a head) is 0.5, and the probability of failure (obtaining a tail) is also 0.5. Let X be the random variable that depicts the number of coin flips required to achieve the initial 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 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 obtaining 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 a fair die up until the initial six shows up.
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 all other number) is 5/6. Let X be the random variable which depicts the number of die rolls required to obtain 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 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 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 theory in probability theory. It is used to model a broad range of practical scenario, such as the count of trials required to get the initial success in different situations.
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