Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is ac crucial branch of mathematics which deals with the study of random occurrence. One of the important theories in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution which models the amount of tests needed to get the first success in a sequence of Bernoulli trials. In this blog, 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 which portrays the number of tests required to accomplish the first success in a sequence of Bernoulli trials. A Bernoulli trial is an experiment which has two viable results, typically referred to as success and failure. Such as flipping a coin is a Bernoulli trial since it can either come up heads (success) or tails (failure).
The geometric distribution is utilized when the trials are independent, meaning that the result of one trial does not affect the result of the upcoming trial. Additionally, the probability of success remains constant throughout all the trials. We can 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 given by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable which depicts the number of test required to get the initial success, k is the count of experiments needed to obtain 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 defined as the anticipated value of the number of trials required 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 likely number of experiments required to obtain the initial success. Such as 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 some primary examples of geometric distribution
Example 1: Tossing a fair coin up until the first head appears.
Suppose we toss an honest coin till the initial head shows up. The probability of success (getting a head) is 0.5, and the probability of failure (getting a tail) is as well as 0.5. Let X be the random variable that represents the count of coin flips required 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 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 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 a fair die till the first six turns up.
Let’s assume we roll a fair die till the initial six appears. The probability of success (achieving a six) is 1/6, and the probability of failure (obtaining any other number) is 5/6. Let X be the irregular variable that depicts the number of die rolls needed to get the first 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 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 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 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 important theory in probability theory. It is used to model a wide array of real-life scenario, for instance the count of experiments required to get the first success in several scenarios.
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