Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is a important department of math which takes up the study of random events. One of the important theories 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 initial success in a series of Bernoulli trials. In this blog, we will explain the geometric distribution, extract its formula, discuss its mean, and give examples.
Meaning of Geometric Distribution
The geometric distribution is a discrete probability distribution which narrates the number of experiments needed to accomplish the first success in a series of Bernoulli trials. A Bernoulli trial is a test which has two likely results, usually referred to as success and failure. For instance, tossing a coin is a Bernoulli trial since it can likewise come up heads (success) or tails (failure).
The geometric distribution is utilized when the trials are independent, which means that the consequence of one test doesn’t affect the result of the upcoming trial. Furthermore, the probability of success remains constant across all the trials. 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 provided by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable that represents the amount of test needed to attain the first success, k is the count of tests needed to obtain 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 anticipated value of the number of experiments 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 a single Bernoulli trial.
The mean is the anticipated count of trials required to get the first success. For example, if the probability of success is 0.5, therefore we expect to get the initial success after two trials on average.
Examples of Geometric Distribution
Here are few primary examples of geometric distribution
Example 1: Tossing a fair coin till the first head turn up.
Imagine we toss a fair coin till the first head appears. The probability of success (getting a head) is 0.5, and the probability of failure (obtaining a tail) is as well as 0.5. Let X be the random variable that represents the count of coin flips needed to get 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 obtaining 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 obtaining 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 initial six appears.
Suppose we roll a fair die until the initial six appears. The probability of success (achieving a six) is 1/6, and the probability of failure (getting any other number) is 5/6. Let X be the irregular variable which depicts the number of die rolls needed to obtain the initial 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 obtaining 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 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 important concept in probability theory. It is utilized to model a broad array of real-world phenomena, for example the number of tests needed to obtain the first success in several situations.
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