Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is a important department of math that takes up the study of random events. One of the important concepts in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution that models the amount of trials needed to obtain the first success in a series of Bernoulli trials. In this article, we will define 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 narrates the number of experiments required to reach the first success in a sequence of Bernoulli trials. A Bernoulli trial is an experiment which has two possible outcomes, usually referred to as success and failure. For example, tossing a coin is a Bernoulli trial since it can either come up heads (success) or tails (failure).
The geometric distribution is utilized when the tests are independent, which means that the outcome of one test does not impact the outcome of the upcoming trial. Furthermore, the chances of success remains constant across all the tests. 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 portrays the amount of test needed to attain the initial success, k is the count of trials 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 described as the anticipated value of the amount of test needed to achieve the first success. The mean is stated in 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 count of experiments needed to get the first success. Such as if the probability of success is 0.5, then we anticipate to obtain the first success following two trials on average.
Examples of Geometric Distribution
Here are handful of primary examples of geometric distribution
Example 1: Flipping a fair coin till the first head shows up.
Let’s assume we flip a fair coin until the first head shows up. The probability of success (obtaining 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 number of coin flips needed to achieve 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 achieving 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 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 forth.
Example 2: Rolling a fair die till the initial six shows up.
Suppose we roll a fair die until the first six shows up. The probability of success (getting 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 count of die rolls required to achieve 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 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 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 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 concept in probability theory. It is utilized to model a broad array of real-life phenomena, such as the count of trials needed to achieve the first success in different situations.
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