Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is ac crucial division of mathematics that deals with the study of random events. One of the crucial ideas in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution that models the amount of experiments required to obtain the initial 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 offer examples.
Definition of Geometric Distribution
The geometric distribution is a discrete probability distribution which portrays the amount of tests required to reach the first success in a succession of Bernoulli trials. A Bernoulli trial is a test that has two likely results, usually indicated to as success and failure. For instance, tossing a coin is a Bernoulli trial because it can either turn out to be heads (success) or tails (failure).
The geometric distribution is utilized when the trials are independent, meaning that the result of one test does not affect the result of the upcoming trial. Additionally, the chances of success remains same across 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 amount of test required to get the initial success, k is the number of trials required 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 defined as the likely value of the amount of trials required to get the initial success. The mean is given by 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 experiments required to achieve the initial success. Such as if the probability of success is 0.5, therefore we anticipate to get the initial success following two trials on average.
Examples of Geometric Distribution
Here are few basic examples of geometric distribution
Example 1: Tossing a fair coin until the first head appears.
Let’s assume we flip a fair coin until the initial head shows up. The probability of success (obtaining a head) is 0.5, and the probability of failure (getting a tail) is also 0.5. Let X be the random variable which portrays the number of coin flips needed to achieve 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 getting 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 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 forth.
Example 2: Rolling an honest die up until the initial six shows up.
Suppose we roll a fair die up until the first six turns 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 that depicts the count of die rolls needed to get the first 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 obtaining the first 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 first 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 wide array of real-world phenomena, for example the number of experiments needed to achieve the initial success in several situations.
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