Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is ac crucial department of math which handles the study of random occurrence. One of the important theories in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution that models the number of experiments needed to get the first success in a series of Bernoulli trials. In this blog, we will define 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 required to accomplish the initial success in a succession of Bernoulli trials. A Bernoulli trial is a test which has two likely outcomes, usually referred to as success and failure. For example, tossing a coin is a Bernoulli trial because it can likewise turn out to be heads (success) or tails (failure).
The geometric distribution is used when the experiments are independent, which means that the outcome of one trial does not affect the outcome of the next test. Furthermore, the chances of success remains unchanged across all the tests. We could 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 which depicts the number of test needed to achieve the initial success, k is the number of trials needed to obtain 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 amount of experiments required to get 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 an individual Bernoulli trial.
The mean is the expected number of tests needed to achieve the initial success. For example, if the probability of success is 0.5, therefore we anticipate to get the first success after two trials on average.
Examples of Geometric Distribution
Here are few primary examples of geometric distribution
Example 1: Tossing a fair coin till the first head appears.
Suppose we flip a fair coin till the initial head turns up. 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 needed 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 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 obtaining 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 a fair die until the first six shows up.
Suppose we roll a fair die until the first six turns up. The probability of success (getting a six) is 1/6, and the probability of failure (getting all other number) is 5/6. Let X be the irregular variable that depicts the number of die rolls required to obtain 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 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 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 forth.
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The geometric distribution is an essential theory in probability theory. It is used to model a broad range of practical phenomena, for example the count of experiments required to achieve the first success in various scenarios.
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