Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is an essential department of math which handles the study of random occurrence. One of the crucial concepts in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution which models the number of experiments needed to obtain the first success in a series of Bernoulli trials. In this blog article, we will define the geometric distribution, extract its formula, discuss its mean, and provide examples.
Meaning of Geometric Distribution
The geometric distribution is a discrete probability distribution that narrates the number of trials required to accomplish the initial success in a succession of Bernoulli trials. A Bernoulli trial is an experiment that has two viable outcomes, typically indicated to as success and failure. For instance, tossing a coin is a Bernoulli trial since it can likewise turn out to be heads (success) or tails (failure).
The geometric distribution is used when the trials are independent, which means that the result of one trial does not impact the result of the upcoming test. Furthermore, the chances of success remains same throughout all the trials. 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 portrays the number of trials required to get the first success, k is the count of trials needed to attain the first success, p is the probability of success in an individual Bernoulli trial, and 1-p is the probability of failure.
Mean of Geometric Distribution:
The mean of the geometric distribution is defined as the expected value of the number of trials 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 anticipated number of trials needed to achieve the first success. For example, if the probability of success is 0.5, therefore we expect to get the first success following two trials on average.
Examples of Geometric Distribution
Here are some primary examples of geometric distribution
Example 1: Flipping a fair coin up until the first head shows up.
Suppose we toss a fair coin till the initial head shows 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 portrays the number of coin flips needed to achieve the initial 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 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 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 an honest die until the initial six turns up.
Suppose we roll an honest die until the first six shows 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 random variable that portrays the number of die rolls needed 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 first 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 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 real-world phenomena, such as the number of tests required to get the initial success in several situations.
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