Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is a important division of math which deals with the study of random occurrence. One of the important concepts in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution which models the amount of tests required to obtain the first success in a secession of Bernoulli trials. In this blog article, we will define the geometric distribution, derive its formula, discuss its mean, and offer examples.
Meaning of Geometric Distribution
The geometric distribution is a discrete probability distribution that describes the amount of trials required to accomplish the first success in a series of Bernoulli trials. A Bernoulli trial is a trial that has two possible results, typically referred to as success and failure. For example, tossing a coin is a Bernoulli trial since it can likewise come up heads (success) or tails (failure).
The geometric distribution is applied when the experiments are independent, which means that the consequence of one trial does not affect the result of the next test. In addition, the chances of success remains constant throughout 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 given by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable that depicts the number of trials needed to attain the initial success, k is the count of experiments needed to achieve 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 explained as the expected 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 a single Bernoulli trial.
The mean is the expected number of tests needed to achieve the initial success. Such as if the probability of success is 0.5, then we expect to attain the first success following two trials on average.
Examples of Geometric Distribution
Here are some primary examples of geometric distribution
Example 1: Tossing a fair coin up until the first head shows up.
Suppose we flip an honest coin till the first head appears. 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 depicts the number of coin flips needed to get the first 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 initial 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 initial 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 up until the first six turns up.
Let’s assume we roll an honest die until the initial six shows up. The probability of success (getting a six) is 1/6, and the probability of failure (achieving any other number) is 5/6. Let X be the random variable which depicts the count of die rolls required to achieve 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 initial 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 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 concept in probability theory. It is utilized to model a wide range of real-world scenario, for instance the number of tests required to obtain the first success in different situations.
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