The Significance of Poisson Distribution in Statistics | Hashtag Statistics
The very first thing a statistician wants to know before diving deep into some analysis is the distribution of given data-set.
How study of distribution of any data-set becomes so dominantly prerequisite for any statistical analysis?
The answer is simple!
For example, if your data-set is Normally distributed, you can easily know what's next using it's different parameters such as Mean and Standard Deviation.
68% of the observations are within one Standard Deviations of the Mean, 95% are within two Standard Deviations, and 99.7% are within three Standard Deviations.
The problem is, in this diverse world, we have diverse data-set too!
Not every data-set can have symmetric distribution with a characteristic of 'bell' shape. Additionally Normal Distribution describes only about continuous data. What if our data is binary or discrete? Well, in this case, we have other distributions e.g. Binomial Distribution, Poisson Distribution.
So, what makes Poisson Distribution unique in the crowd?
- The Poisson Distribution represents discrete counts of occurrences in a continuous time/number of events.
- It is not symmetrical; it is skewed toward the infinity end.
Well, leave any statistical term for a moment! Let's jump into a real world!
- Number of earthquake in a given period of time
- Number of birth/death in a given period of time
- Number of marriages/divorces in a given period of time
- Number of suicides in a given period of time
- Number of heart-attack patient arrival in clinic in a given period of time
- Number of bankruptcies that are filed in a given period of time
- Number of bugs per byte of code
What are the most common characteristics have you observed in the given example?
1. "given period of time"
2. The event seems to be very rare, isn't it?
Long story short, Poisson Distribution is a
Law of rare events!
or
When the number of events tend to infinite and the probability tends to zero!
The connection between the Poisson and Binomial Distributions
- The Poisson Distribution is a limiting case of a Binomial Distribution when the number of trials, n, gets very large and p, the probability of success, is very small.
- Binomial Probability of being x successes in n trials
P(x) = nCx p^x q^(n−x)
p = μ/n
q = 1 − μ/n
P(x) = nCx (μ/n)^x (1−μ/n)^(n−x)
i.e.
lim n→∞ P(x)=e^(−μ) μ^x / x!
In short,
P(x; μ) = (e^(-μ) * μ^(x)) / x!
Where,
μ = Average number of events
e = Euler's constant ~ 2.71828
x = Number of success for the event
Example:
- In an automobile manufacturing company, the probability that an electric motor is defective is 0.02. What is the probability that a sample of 400 electric motors will contain exactly 5 defective motors?
The average number of defectives in 400 motors is μ = 0.02 × 400 = 8
The probability of getting 5 defectives is:
P(X)= ((e^(-8))*8^5)/5!
= 0.00033546262 * 32768 / 120
= 0.09160
So, like Normal Distribution, what are the Mean and Variance of Poisson Distribution?
If μ is the average number of successes occurring in a given time interval then the Mean and the Variance of the Poisson distribution are both equal to μ.
Mean = μ
and
Variance = σ^2 = μ
From the above discussion, it can be seen that in Poisson distribution, only one parameter, μ is needed to determine the probability of an event.
The Poisson Distribution is applicable where:
- The event is something that can be counted in whole numbers or discrete.
- The occurrences of events are independent.
- The average frequency of occurrence for the time period is known.
- It is possible to count how many events have been occurred in a given period of time.
I was reading about Bernoulli distribution that also seems to be so significant but is relatively complex to read.
ReplyDeleteAmazing article there!
ReplyDeleteThe significance of Poisson lies in his profound contributions to probability theory and statistics. Slow Device Streaming His work on the Poisson distribution and the Poisson process is fundamental in modeling random events and phenomena.
ReplyDeleteThe article gives a useful introduction to why understanding the distribution of a dataset is an important starting point in statistical analysis. The comparison between Normal, Binomial, and Poisson distributions makes it clear that different types of data require different probability models, particularly when dealing with discrete counts and events occurring over a given period.
ReplyDeleteThe explanation of Poisson Distribution is especially useful because it connects the mathematical concept with practical examples such as earthquakes, births, deaths, patient arrivals, bankruptcies, and software bugs. This kind of connection between probability theory and real-world data is an important part of Data Analytics Course, where understanding the characteristics of data helps determine which statistical techniques are appropriate.
I also found the distinction between continuous and discrete data particularly helpful. The article explains why Normal Distribution cannot simply be applied to every dataset and introduces Poisson Distribution as an alternative for counting events. Such probability concepts provide important foundations for Machine Learning Training, where understanding the underlying distribution of data can influence how models and statistical assumptions are interpreted.
ReplyDeleteThe real-world examples make the discussion easier to understand because they demonstrate the common characteristics of Poisson-type problems: a defined period or interval and the counting of events occurring within that interval. These concepts can also be useful when developing analytical solutions and experimental studies, which are common areas of work in Data Science Projects For Final Year.
ReplyDeleteOverall, the article provides a straightforward explanation of why Poisson Distribution is useful for modeling event counts and why identifying the characteristics of a dataset should come before selecting a statistical method. The examples provide a practical starting point for students exploring probability, statistics, and applied computational analysis, including work related to Machine Learning Projects For Final Year.
ReplyDelete