Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is ac crucial branch of mathematics that handles the study of random events. One of the essential concepts in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution that models the amount of experiments needed to obtain the first success in a secession of Bernoulli trials. In this blog, we will talk about the geometric distribution, derive its formula, discuss its mean, and provide examples.
Meaning of Geometric Distribution
The geometric distribution is a discrete probability distribution that describes the amount of tests required to reach the first success in a succession of Bernoulli trials. A Bernoulli trial is a test which has two possible outcomes, usually referred to as success and failure. For example, flipping a coin is a Bernoulli trial because it can likewise turn out to be heads (success) or tails (failure).
The geometric distribution is utilized when the experiments are independent, which means that the outcome of one trial doesn’t affect the outcome of the next trial. In addition, 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 which depicts the number of trials required to achieve the initial success, k is the number of tests required to achieve the initial 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 expected value of the amount of trials needed to get 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 anticipated number of tests needed to obtain the initial success. Such as if the probability of success is 0.5, therefore we expect to get the first success following two trials on average.
Examples of Geometric Distribution
Here are some primary examples of geometric distribution
Example 1: Flipping a fair coin till the first head appears.
Imagine we flip a fair coin till the first head turns 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 depicts the count of coin flips required to obtain 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 achieving 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 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 getting 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 till the initial six appears.
Suppose we roll a fair die up until the first six turns up. The probability of success (getting a six) is 1/6, and the probability of failure (achieving any other number) is 5/6. Let X be the irregular variable that portrays the count of die rolls required 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 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 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 getting 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 utilized to model a wide array of real-world phenomena, for instance the count of trials needed to obtain the initial success in different situations.
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