Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is an essential department of math which takes up the study of random events. One of the essential theories in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution that models the number of experiments required to obtain the initial success in a secession of Bernoulli trials. In this blog, we will define the geometric distribution, extract 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 tests needed to accomplish the initial success in a sequence of Bernoulli trials. A Bernoulli trial is an experiment that has two possible outcomes, usually indicated to as success and failure. Such as tossing a coin is a Bernoulli trial because it can likewise come up heads (success) or tails (failure).
The geometric distribution is applied when the trials are independent, which means that the result of one trial doesn’t affect the result of the upcoming test. In addition, the chances of success remains same 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 specified by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable which portrays the number of trials needed to get the first success, k is the count of experiments needed to achieve 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 explained as the anticipated value of the number of test needed to get the first success. The mean is given by 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 trials needed to achieve the initial success. For instance, if the probability of success is 0.5, then we anticipate to obtain the first success after two trials on average.
Examples of Geometric Distribution
Here are few basic examples of geometric distribution
Example 1: Tossing a fair coin till the first head turn up.
Let’s assume we toss an honest coin until 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 which portrays the number of coin flips required 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 initial head on the third flip is:
P(X = 3) = 0.5^(3-1) * 0.5 = 0.125
And so on.
Example 2: Rolling a fair die up until the first six turns up.
Let’s assume we roll an honest die till 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 random variable which represents the number of die rolls needed to obtain the initial six. The PMF of X is stated as:
P(X = k) = (1 - 1/6)^(k-1) * 1/6 = (5/6)^(k-1) * 1/6
For k = 1, the probability of getting the initial six on the first 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 achieving the initial 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 an essential concept in probability theory. It is used to model a broad array of real-life phenomena, such as the number of tests required to obtain the initial success in various situations.
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