Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is a important branch of mathematics that deals with 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 amount of tests required to get the initial success in a series of Bernoulli trials. In this blog article, we will explain the geometric distribution, extract its formula, discuss its mean, and offer examples.
Meaning of Geometric Distribution
The geometric distribution is a discrete probability distribution that portrays the amount of tests required to reach the initial success in a series of Bernoulli trials. A Bernoulli trial is a trial that has two likely results, usually referred to as success and failure. For instance, tossing a coin is a Bernoulli trial since it can either turn out to be heads (success) or tails (failure).
The geometric distribution is used when the experiments are independent, which means that the consequence of one test doesn’t affect the outcome of the upcoming test. Furthermore, the probability of success remains unchanged throughout all the tests. 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 given by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable that portrays the number of test needed to achieve the first success, k is the number of trials 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 explained as the expected value of the number of trials required 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 a single Bernoulli trial.
The mean is the likely count of tests needed to obtain the initial success. For example, if the probability of success is 0.5, then we anticipate to get the first success after two trials on average.
Examples of Geometric Distribution
Here are some basic examples of geometric distribution
Example 1: Flipping a fair coin until the first head appears.
Let’s assume we flip a fair coin until the first head shows up. The probability of success (getting a head) is 0.5, and the probability of failure (obtaining a tail) is also 0.5. Let X be the random variable which portrays the count of coin flips needed to achieve the initial 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 initial 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 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.
Let’s assume we roll a fair die up until the initial six appears. The probability of success (obtaining a six) is 1/6, and the probability of failure (achieving any other number) is 5/6. Let X be the irregular variable that represents the number of die rolls needed to achieve 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 initial six on the initial roll is:
P(X = 1) = (5/6)^(1-1) * 1/6 = 1/6
For k = 2, the probability of achieving 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 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 crucial theory in probability theory. It is applied to model a broad array of real-life phenomena, such as the number of trials needed to obtain the initial success in several situations.
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