Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is a important 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 experiments required to obtain the initial success in a series of Bernoulli trials. In this blog, we will talk about the geometric distribution, extract its formula, discuss its mean, and provide examples.
Meaning of Geometric Distribution
The geometric distribution is a discrete probability distribution which portrays the number of tests required to achieve the initial success in a series of Bernoulli trials. A Bernoulli trial is a trial that has two possible outcomes, usually referred to as success and failure. For instance, tossing a coin is a Bernoulli trial because it can likewise come up heads (success) or tails (failure).
The geometric distribution is utilized when the trials are independent, which means that the outcome of one experiment doesn’t affect the outcome of the next trial. In addition, the chances of success remains constant across all the trials. We could 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 specified by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable which portrays the amount of trials required to achieve the initial success, k is the number of tests required 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 explained as the anticipated value of the number of test needed to get 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 needed to achieve the initial success. For instance, if the probability of success is 0.5, then we expect to obtain the first success after two trials on average.
Examples of Geometric Distribution
Here are some essential examples of geometric distribution
Example 1: Flipping a fair coin up until the first head shows up.
Suppose we flip a fair coin till the first head turns 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 which portrays the count of coin flips required to achieve the first head. The PMF of X is provided 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 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 first head on the third flip is:
P(X = 3) = 0.5^(3-1) * 0.5 = 0.125
And so on.
Example 2: Rolling an honest die till the initial six turns up.
Suppose we roll an honest die up until the initial six appears. The probability of success (obtaining a six) is 1/6, and the probability of failure (obtaining any other number) is 5/6. Let X be the irregular variable which portrays the count of die rolls needed 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 initial six on the first 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 achieving 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 important theory in probability theory. It is used to model a wide range of real-world phenomena, such as the number of tests needed to obtain the first success in various scenarios.
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