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 important theories in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution that models the amount of experiments required to get the initial success in a sequence of Bernoulli trials. In this blog article, we will talk about the geometric distribution, extract its formula, discuss its mean, and provide examples.
Explanation of Geometric Distribution
The geometric distribution is a discrete probability distribution which describes the amount of experiments needed to reach the first success in a succession of Bernoulli trials. A Bernoulli trial is an experiment which has two likely results, generally referred to as success and failure. For instance, flipping a coin is a Bernoulli trial because it can likewise come up heads (success) or tails (failure).
The geometric distribution is used when the experiments are independent, meaning that the consequence of one trial doesn’t affect the result of the next trial. Furthermore, the probability of success remains constant across all the tests. We could indicate 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 which portrays the amount of test needed to get the first success, k is the count of tests required to obtain 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 expected value of the number of experiments 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 expected number of trials required to achieve the initial success. Such as if the probability of success is 0.5, then we anticipate to attain the first success after two trials on average.
Examples of Geometric Distribution
Here are few essential examples of geometric distribution
Example 1: Flipping a fair coin till the first head turn up.
Imagine we flip an honest coin until the initial head appears. The probability of success (getting a head) is 0.5, and the probability of failure (getting a tail) is also 0.5. Let X be the random variable which represents the count of coin flips required to obtain the first 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 getting 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 first 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 up until the initial six turns up.
Let’s assume we roll an honest die until the initial six appears. The probability of success (getting a six) is 1/6, and the probability of failure (getting all other number) is 5/6. Let X be the random variable which represents the count of die rolls needed to achieve 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 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 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 forth.
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The geometric distribution is a important theory in probability theory. It is utilized to model a broad range of practical scenario, for example the count of experiments required to obtain the first success in several situations.
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