Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is ac crucial department of mathematics which handles the study of random events. One of the crucial theories in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution that models the number of trials needed to obtain the initial success in a secession of Bernoulli trials. In this article, we will explain the geometric distribution, derive its formula, discuss its mean, and provide examples.
Explanation of Geometric Distribution
The geometric distribution is a discrete probability distribution that narrates the number of experiments required to accomplish the initial success in a succession of Bernoulli trials. A Bernoulli trial is a trial that has two viable outcomes, generally indicated to as success and failure. For example, flipping a coin is a Bernoulli trial since it can either turn out to be heads (success) or tails (failure).
The geometric distribution is applied when the tests are independent, meaning that the consequence of one experiment does not impact the outcome of the upcoming trial. Furthermore, the chances of success remains constant throughout all the tests. We can 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 that represents the number of trials required to achieve the first success, k is the count of trials needed 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 described as the anticipated value of the number of experiments 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 count of trials required to achieve the initial success. For instance, if the probability of success is 0.5, therefore we expect to attain the first success after 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.
Let’s assume we toss a fair coin until the initial head turns up. The probability of success (obtaining a head) is 0.5, and the probability of failure (getting a tail) is also 0.5. Let X be the random variable that represents the count of coin flips needed to achieve 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 obtaining 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 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 forth.
Example 2: Rolling a fair die until the initial six appears.
Suppose we roll a fair die up until the initial six appears. The probability of success (getting a six) is 1/6, and the probability of failure (getting any other number) is 5/6. Let X be the irregular variable that represents the number of die rolls needed to achieve 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 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 achieving 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 crucial theory in probability theory. It is utilized to model a wide array of practical phenomena, for example the number of experiments required to obtain the initial success in different situations.
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