Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is an essential branch of mathematics which handles the study of random events. One of the important concepts in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution which models the number of experiments required to get the initial success in a sequence of Bernoulli trials. In this article, we will define the geometric distribution, extract its formula, discuss its mean, and offer examples.
Explanation of Geometric Distribution
The geometric distribution is a discrete probability distribution which narrates the amount of tests required to achieve the first success in a series of Bernoulli trials. A Bernoulli trial is a test which has two likely outcomes, generally indicated to as success and failure. For example, tossing a coin is a Bernoulli trial since it can likewise come up heads (success) or tails (failure).
The geometric distribution is utilized when the tests are independent, which means that the result of one experiment does not affect the outcome of the upcoming test. Furthermore, the chances of success remains same across 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 provided by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable which portrays the amount of trials needed to get the first success, k is the number of experiments 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 likely 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 anticipated number of trials required to obtain the first success. For instance, if the probability of success is 0.5, then we anticipate to attain the initial 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 turn up.
Let’s assume we flip a fair coin until the first head turns up. 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 that portrays the count of coin flips needed to achieve the initial 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 getting the first 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 appears.
Suppose we roll an honest die until the initial six turns up. 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 represents 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 first roll is:
P(X = 1) = (5/6)^(1-1) * 1/6 = 1/6
For k = 2, the probability of obtaining 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 an essential concept in probability theory. It is used to model a wide array of real-world phenomena, such as the count of tests required to get the first success in different scenarios.
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