Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is a important branch of mathematics which deals with the study of random events. One of the essential theories in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution that models the amount of experiments required to get the first success in a secession of Bernoulli trials. In this blog article, we will talk about the geometric distribution, derive its formula, discuss its mean, and give examples.
Definition of Geometric Distribution
The geometric distribution is a discrete probability distribution which describes the number of experiments required to reach the first success in a series of Bernoulli trials. A Bernoulli trial is a test which has two viable outcomes, typically referred to as success and failure. Such as 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, meaning that the outcome of one test doesn’t impact the outcome of the upcoming trial. Additionally, the chances of success remains same across all the trials. We can 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 which depicts the amount of trials required to attain the initial success, k is the count of tests needed to achieve 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 described as the anticipated value of the number of test needed to obtain 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 expected count of tests required to get the first success. For example, if the probability of success is 0.5, then we expect to get the first success following two trials on average.
Examples of Geometric Distribution
Here are some essential examples of geometric distribution
Example 1: Tossing a fair coin till the first head shows up.
Let’s assume we toss an honest coin until the first 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 portrays the number of coin flips required to obtain the initial head. The PMF of X is given by:
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 initial 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 shows up.
Suppose we roll an honest die up until the initial six turns up. The probability of success (achieving a six) is 1/6, and the probability of failure (getting all other number) is 5/6. Let X be the random variable that represents the count of die rolls required to obtain the initial 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 getting 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 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 forth.
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The geometric distribution is a important concept in probability theory. It is used to model a broad range of real-life phenomena, for example the count of trials needed to obtain the first success in different situations.
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