Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is a important department of mathematics which handles the study of random events. One of the important theories 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 secession of Bernoulli trials. In this article, we will define the geometric distribution, extract its formula, discuss its mean, and give examples.
Explanation of Geometric Distribution
The geometric distribution is a discrete probability distribution which describes the amount of experiments required to achieve the initial success in a series of Bernoulli trials. A Bernoulli trial is a trial which has two possible outcomes, usually referred to as success and failure. Such as flipping a coin is a Bernoulli trial because it can either come up heads (success) or tails (failure).
The geometric distribution is utilized when the trials are independent, which means that the result of one trial does not affect the result of the next trial. Furthermore, the chances of success remains unchanged throughout 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 test required to achieve the initial success, k is the count of trials needed to attain 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 amount of trials needed to obtain the first 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 tests required to obtain the first success. For instance, if the probability of success is 0.5, therefore we expect to get the first success following two trials on average.
Examples of Geometric Distribution
Here are few essential examples of geometric distribution
Example 1: Tossing a fair coin till the first head turn up.
Suppose we toss an honest coin till the initial head appears. 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 which portrays the count of coin flips required to obtain the first 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 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 first 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 first six shows up.
Let’s assume we roll a fair die until the initial six appears. 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 which depicts the count of die rolls needed to obtain the first 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 obtaining the initial six on the initial 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 first 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 crucial concept in probability theory. It is used to model a broad array of practical phenomena, such as the number of experiments required to obtain the initial success in various situations.
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