Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is ac crucial department of math which handles the study of random occurrence. One of the essential concepts in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution that models the number of trials required to get the initial success in a series of Bernoulli trials. In this blog article, we will define the geometric distribution, extract its formula, discuss its mean, and provide examples.
Definition of Geometric Distribution
The geometric distribution is a discrete probability distribution which describes the number of trials required to achieve the first success in a succession of Bernoulli trials. A Bernoulli trial is a test which has two viable results, usually indicated to as success and failure. Such as 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 experiment doesn’t impact the result of the upcoming test. Furthermore, the probability of success remains same across 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 provided by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable that depicts the number of test needed to achieve the first success, k is the number of trials needed to obtain 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 expected value of the number of experiments required to achieve 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 likely number of trials needed to achieve the initial success. For instance, if the probability of success is 0.5, then we expect to attain the initial success after two trials on average.
Examples of Geometric Distribution
Here are handful of primary examples of geometric distribution
Example 1: Tossing a fair coin until the first head turn up.
Suppose we toss a fair coin till the first 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 depicts the number of coin flips needed to achieve 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 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 getting 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 forth.
Example 2: Rolling a fair die till the initial six shows up.
Suppose we roll a fair die until the initial six turns up. The probability of success (achieving a six) is 1/6, and the probability of failure (achieving any other number) is 5/6. Let X be the irregular variable that depicts the number of die rolls required to get 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 achieving 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 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 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 theory in probability theory. It is used to model a wide range of practical phenomena, such as the number of experiments required to get the first success in various scenarios.
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