Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is a important division of mathematics that handles the study of random occurrence. One of the essential ideas in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution which models the number of experiments required to obtain the initial success in a sequence of Bernoulli trials. In this blog, we will explain the geometric distribution, derive 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 experiments needed to accomplish the initial success in a series of Bernoulli trials. A Bernoulli trial is a trial which has two possible results, typically referred to as success and failure. Such as tossing a coin is a Bernoulli trial because it can likewise turn out to be heads (success) or tails (failure).
The geometric distribution is utilized when the trials are independent, meaning that the consequence of one trial does not affect the outcome of the upcoming test. Furthermore, the chances of success remains same across 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 represents the number of trials needed to attain the first success, k is the number of experiments needed to attain 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 defined as the likely value of the amount of test needed 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 anticipated number of trials needed to achieve the first success. For instance, if the probability of success is 0.5, then we expect to get the first success after two trials on average.
Examples of Geometric Distribution
Here are handful of primary examples of geometric distribution
Example 1: Flipping a fair coin till the first head appears.
Imagine we toss a fair coin till the initial head appears. 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 represents the count of coin flips required to achieve the initial head. The PMF of X is given by:
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 forth.
Example 2: Rolling an honest die until the initial six shows up.
Suppose we roll a fair 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 random variable which represents the number of die rolls needed to get the first six. The PMF of X is stated 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 first roll is:
P(X = 1) = (5/6)^(1-1) * 1/6 = 1/6
For k = 2, the probability of getting 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 getting 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 broad range of real-world phenomena, for instance the count of trials needed to obtain the first success in several scenarios.
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