Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is a important division of math that handles the study of random occurrence. One of the crucial theories in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution that models the amount of trials required to get the initial success in a secession of Bernoulli trials. In this blog article, we will talk about the geometric distribution, derive its formula, discuss its mean, and give examples.
Meaning of Geometric Distribution
The geometric distribution is a discrete probability distribution which portrays the amount of experiments needed to achieve the initial success in a series of Bernoulli trials. A Bernoulli trial is a trial which has two viable results, typically referred to as success and failure. For instance, tossing 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 tests are independent, meaning that the outcome of one trial does not impact the outcome of the next test. Additionally, the probability of success remains same across all the tests. We can denote 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 number of test required to attain the first success, k is the number of trials required to achieve the first success, p is the probability of success in a single Bernoulli trial, and 1-p is the probability of failure.
Mean of Geometric Distribution:
The mean of the geometric distribution is defined as the expected value of the number of trials required to obtain 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 an individual Bernoulli trial.
The mean is the anticipated count of experiments required to obtain the initial success. Such as if the probability of success is 0.5, therefore we expect to attain the initial success following two trials on average.
Examples of Geometric Distribution
Here are some primary examples of geometric distribution
Example 1: Tossing a fair coin until the first head turn up.
Suppose we flip an honest coin till the initial head shows up. The probability of success (obtaining a head) is 0.5, and the probability of failure (obtaining a tail) is as well as 0.5. Let X be the random variable that depicts the number of coin flips needed to get 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 initial 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 obtaining 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 a fair die till the first six turns up.
Suppose we roll a fair die up until the first six shows up. The probability of success (obtaining a six) is 1/6, and the probability of failure (achieving any other number) is 5/6. Let X be the irregular variable that represents the number of die rolls needed to get the initial six. The PMF of X is provided as:
P(X = k) = (1 - 1/6)^(k-1) * 1/6 = (5/6)^(k-1) * 1/6
For k = 1, the probability of achieving 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 an essential concept in probability theory. It is utilized to model a broad range of practical phenomena, for example the number of trials needed to achieve the initial success in various situations.
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