Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is an essential department of mathematics which handles the study of random occurrence. One of the essential ideas in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution which models the number of experiments needed to get the initial success in a secession of Bernoulli trials. In this blog, we will talk about the geometric distribution, extract its formula, discuss its mean, and give examples.
Explanation of Geometric Distribution
The geometric distribution is a discrete probability distribution which describes the amount of tests required to accomplish the first success in a series of Bernoulli trials. A Bernoulli trial is a trial which has two possible results, generally indicated to as success and failure. For example, flipping a coin is a Bernoulli trial because it can likewise turn out to be heads (success) or tails (failure).
The geometric distribution is used when the trials are independent, meaning that the result of one experiment doesn’t impact the result of the next test. Additionally, the chances of success remains constant 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 given by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable which portrays the number of test needed to achieve the first success, k is the count of tests needed 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 defined as the likely value of the amount of test needed to get 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 expected count of trials needed to get the initial success. For example, if the probability of success is 0.5, therefore we expect to obtain the first success after two trials on average.
Examples of Geometric Distribution
Here are some primary examples of geometric distribution
Example 1: Tossing a fair coin up until the first head turn up.
Suppose we flip an honest coin till the initial 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 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 obtaining 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 achieving 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 an honest die till the initial six turns up.
Let’s assume we roll an honest die up until the initial six turns up. The probability of success (getting a six) is 1/6, and the probability of failure (achieving all other number) is 5/6. Let X be the random variable that depicts the number of die rolls needed to get 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 initial 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 a important concept in probability theory. It is applied to model a broad array of real-life phenomena, such as the count of tests required to obtain the first success in different situations.
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