Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is a important division of math 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 required to get the initial success in a secession of Bernoulli trials. In this blog, we will talk about the geometric distribution, extract its formula, discuss its mean, and provide examples.
Definition of Geometric Distribution
The geometric distribution is a discrete probability distribution which describes the amount of tests required to reach the first success in a sequence of Bernoulli trials. A Bernoulli trial is an experiment which has two likely outcomes, usually referred to as success and failure. For example, flipping a coin is a Bernoulli trial since it can likewise turn out to be heads (success) or tails (failure).
The geometric distribution is applied when the experiments are independent, which means that the consequence of one test does not impact the outcome of the next test. Furthermore, the chances of success remains constant across all the trials. 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 provided by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable that portrays the number of test needed to attain the first success, k is the number of tests needed to obtain the initial 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 expected value of the number of experiments needed to get the initial success. The mean is stated in 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 experiments required to obtain the initial success. For example, if the probability of success is 0.5, then we anticipate to get the initial success following two trials on average.
Examples of Geometric Distribution
Here are handful of essential examples of geometric distribution
Example 1: Flipping a fair coin till the first head appears.
Suppose we flip a fair coin until the initial head shows up. The probability of success (getting 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 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 first 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 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 turns up.
Suppose we roll an honest die up until the initial six shows up. The probability of success (obtaining a six) is 1/6, and the probability of failure (obtaining all other number) is 5/6. Let X be the irregular variable which represents the count of die rolls required 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 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 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 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 theory in probability theory. It is utilized to model a wide array of practical scenario, for instance the count of experiments required to obtain the first success in various scenarios.
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