Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is ac crucial division of mathematics which handles the study of random events. One of the crucial concepts in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution which models the amount of trials needed to obtain the first success in a series of Bernoulli trials. In this blog article, we will define the geometric distribution, extract its formula, discuss its mean, and give examples.
Meaning of Geometric Distribution
The geometric distribution is a discrete probability distribution that describes the number of experiments required to achieve the initial success in a sequence of Bernoulli trials. A Bernoulli trial is a trial which has two viable results, generally referred to as success and failure. Such as tossing a coin is a Bernoulli trial because it can likewise come up heads (success) or tails (failure).
The geometric distribution is applied when the trials are independent, which means that the consequence of one trial does not impact the result of the next test. Additionally, the chances of success remains same throughout all the trials. 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 specified by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable which depicts the number of test needed to achieve the initial success, k is the count of tests 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 likely value of the number of experiments required to achieve 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 expected number of trials needed to obtain the first success. Such as if the probability of success is 0.5, therefore we expect to obtain the initial success following two trials on average.
Examples of Geometric Distribution
Here are some basic examples of geometric distribution
Example 1: Flipping a fair coin up until the first head shows up.
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 depicts the count of coin flips needed to obtain 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 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 obtaining 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 up until the first six turns up.
Let’s assume we roll a fair die till the first six shows up. The probability of success (getting a six) is 1/6, and the probability of failure (obtaining all other number) is 5/6. Let X be the random variable that represents the number of die rolls needed to obtain 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 getting the first six on the initial roll is:
P(X = 1) = (5/6)^(1-1) * 1/6 = 1/6
For k = 2, the probability of obtaining 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 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 concept in probability theory. It is utilized to model a broad range of real-life phenomena, such as the number of tests required to achieve the initial success in several situations.
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