Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is ac crucial department of mathematics that takes up the study of random occurrence. 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 tests required to obtain the initial success in a secession 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 portrays the number of experiments needed to reach the first success in a sequence of Bernoulli trials. A Bernoulli trial is a trial which has two possible outcomes, typically referred to as success and failure. For example, 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 tests are independent, meaning that the outcome of one trial doesn’t affect the outcome of the next test. In addition, the probability of success remains same across all the trials. 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 depicts the number of test required to achieve the initial success, k is the count of trials 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 explained as the expected value of the amount of test needed to obtain 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 likely number of trials required to get the first success. For instance, 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 few primary examples of geometric distribution
Example 1: Flipping a fair coin up until the first head shows up.
Suppose we flip an honest coin till the first head turns 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 depicts 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 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 achieving 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 until the initial six shows up.
Suppose we roll an honest die till the initial six shows 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 needed to achieve 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 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 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 obtaining the initial 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 concept in probability theory. It is utilized to model a wide array of real-life phenomena, such as the count of experiments needed to obtain the first success in different scenarios.
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