# Poisson Distribution Intuition (and derivation) (2022)

## When to use a Poisson Distribution

Before setting the parameter λ and plugging it into the formula, let’s pause a second and ask a question.

Why did Poisson have to invent the Poisson Distribution?

Why does this distribution exist (= why did he invent this)?

When should Poisson be used for modeling?

To predict the # of events occurring in the future!

More formally, to predict the probability of a given number of events occurring in a fixed interval of time.

If you’ve ever sold something, this “event” can be defined, for example, as a customer purchasing something from you (the moment of truth, not just browsing). It can be how many visitors you get on your website a day, how many clicks your ads get for the next month, how many phone calls you get during your shift, or even how many people will die from a fatal disease next year, etc.

Below is an example of how I’d use Poisson in real life.

(Video) Poisson distribution derivation. Intuitive example.

Every week, on average, 17 people clap for my blog post. I’d like to predict the # of ppl who would clap next week because I get paid weekly by those numbers.What is the probability that exactly 20 people (or 10, 30, 50, etc.) will clap for the blog post next week?

One way to solve this would be to start with the number of reads. Each person who reads the blog has some probability that they will really like it and clap.

This is a classic job for the binomial distribution, since we are calculating the probability of the number of successful events (claps).

A binomial random variable is the number of successes x in n repeated trials. And we assume the probability of success p is constant over each trial.

However, here we are given only one piece of information — 17 ppl/week, which is a “rate” (the average # of successes per week, or the expected value of x). We don’t know anything about the clapping probability p, nor the number of blog visitors n.

Therefore, we need a little more information to tackle this problem. What more do we need to frame this probability as a binomial problem? We need two things: the probability of success (claps) p & the number of trials (visitors) n.

Let’s get them from the past data.

These are stats for 1 year. A total of 59k people read my blog. Out of 59k people, 888 of them clapped.

Therefore, the # of people who read my blog per week (n) is 59k/52 = 1134. The # of people who clapped per week (x) is 888/52 =17.

# of people who read per week (n) = 59k/52 = 1134# of people who clap per week (x) = 888/52 = 17Success probability (p) : 888/59k = 0.015 = 1.5%

## Using the Binomial PMF, what is the probability thatI’ll get exactly 20 successes (20 people who clap) next week?

<Binomial Probability for different x’s>╔══════╦════════════════╗║ x ║ Binomial P(X=x)║╠══════╬════════════════╣║ 10 ║ 0.02250 ║║ 17 ║ 0.09701 ║ 🡒 The average rate has the highest P!║ 20 ║ 0.06962 ║ 🡒 Nice. 20 is also quite Likely!║ 30 ║ 0.00121 ║║ 40 ║ < 0.000001 ║ 🡒 Well, I guess I won’t get 40 claps..╚══════╩════════════════╝

We just solved the problem with a binomial distribution.

Then, what is Poisson for? What are the things that only Poisson can do, but Binomial can’t?

a) A binomial random variable is “BI-nary” — 0 or 1.

In the above example, we have 17 ppl/wk who clapped. This means 17/7 = 2.4 people clapped per day, and 17/(7*24) = 0.1 people clapping per hour.

If we model the success probability by hour (0.1 people/hr) using the binomial random variable, this means most of the hours get zero claps but some hours will get exactly 1 clap. However, it is also very possible that certain hours will get more than 1 clap (2, 3, 5 claps, etc.)

The problem with binomial is that it CANNOT contain more than 1 event in the unit of time (in this case, 1 hr is the unit time). The unit of time can only have 0 or 1 event.

Then, how about dividing 1 hour into 60 minutes, and make unit time smaller, for example, a minute? Then 1 hour can contain multiple events. (Still, one minute will contain exactly one or zero events.)

Is our problem solved now?

Kind of. But what if, during that one minute, we get multiple claps? (i.e. someone shared your blog post on Twitter and the traffic spiked at that minute.) Then what? We can divide a minute into seconds. Then our time unit becomes a second and again a minute can contain multiple events. But this binary container problem will always exist for ever-smaller time units.

The idea is, we can make the Binomial random variable handle multiple events by dividing a unit time into smaller units. By using smaller divisions, we can make the original unit time contain more than one event.

Mathematically, this means n → ∞.
Since we assume the rate is fixed, we must have p → 0. Because otherwise, n*p, which is the number of events, will blow up.

(Video) The Poisson Distribution: Derivation

Using the limit, the unit times are now infinitesimal. We no longer have to worry about more than one event occurring within the same unit time. And this is how we derive Poisson distribution.

b) In the Binomial distribution, the # of trials (n) should be known beforehand.

If you use Binomial, you cannot calculate the success probability only with the rate (i.e. 17 ppl/week). You need “more info” (n & p) in order to use the binomial PMF.
The Poisson Distribution, on the other hand, doesn’t require you to know n or p. We are assuming n is infinitely large and p is infinitesimal. The only parameter of the Poisson distribution is the rate λ (the expected value of x). In real life, only knowing the rate (i.e., during 2pm~4pm, I received 3 phone calls) is much more common than knowing both n & p.

Now you know where each component λ^k , k! and e^-λ come from!

Finally, we only need to show that the multiplication of the first two terms n!/((n-k)!*n^k) is 1 when n approaches infinity.

It is 1.

We got the Poisson Formula!

(Video) Proof that the Binomial Distribution tends to the Poisson Distribution

Now the Wikipedia explanation starts making sense.

Plug your own data into the formula and see if P(x) makes sense to you!

Below is mine.

< Comparison between Binomial & Poisson >╔══════╦═══════════════════╦═══════════════════════╗║ k ║ Binomial P(X=k) ║ Poisson P(X=k;λ=17) ║╠══════╬═══════════════════╬═══════════════════════╣║ 10 ║ 0.02250 ║ 0.02300 ║║ 17 ║ 0.09701 ║ 0.09628 ║║ 20 ║ 0.06962 ║ 0.07595 ║║ 30 ║ 0.00121 ║ 0.00340 ║║ 40 ║ < 0.000001 ║ < 0.000001 ║╚══════╩═══════════════════╩═══════════════════════╝* You can calculate both easily here:Binomial: https://stattrek.com/online-calculator/binomial.aspxPoisson : https://stattrek.com/online-calculator/poisson.aspx

A few things to note:

1. Even though the Poisson distribution models rare events, the rate λ can be any number. It doesn’t always have to be small.
2. The Poisson Distribution is asymmetric — it is always skewed toward the right. Because it is inhibited by the zero occurrence barrier (there is no such thing as “minus one” clap) on the left and it is unlimited on the other side.
3. As λ becomes bigger, the graph looks more like a normal distribution.

4. The Poisson Model Assumptions

a. The average rate of events per unit time is constant.
This means the number of people who visit your blog per hour might not follow a Poisson Distribution, because the hourly rate is not constant (higher rate during the daytime, lower rate during the nighttime). Using monthly rate for consumer/biological data would be just an approximation as well, since the seasonality effect is non-trivial in that domain.

b. Events are independent.
The arrivals of your blog visitors might not always be independent. For example, sometimes a large number of visitors come in a group because someone popular mentioned your blog, or your blog got featured on Medium’s first page, etc. The number of earthquakes per year in a country also might not follow a Poisson Distribution if one large earthquake increases the probability of aftershocks.

5. Relationship between a Poisson and an Exponential distribution

(Video) Lecture 11: The Poisson distribution | Statistics 110

If the number of events per unit time follows a Poisson distribution, then the amount of time between events follows the exponential distribution. The Poisson distribution is discrete and the exponential distribution is continuous, yet the two distributions are closely related.

Let’s go deeper: Exponential Distribution Intuition

If you like my post, could you please clap? It gives me motivation to write more. :)

## FAQs

### How is the Poisson distribution derived? ›

The Poisson distribution is a limiting case of the binomial distribution which arises when the number of trials n increases indefinitely whilst the product μ = np, which is the expected value of the number of successes from the trials, remains constant.

### What is Poisson distribution with example? ›

A Poisson distribution is a discrete probability distribution. It gives the probability of an event happening a certain number of times (k) within a given interval of time or space. The Poisson distribution has only one parameter, λ (lambda), which is the mean number of events.

### What are the 3 conditions for a Poisson distribution? ›

Poisson Process Criteria

Events are independent of each other. The occurrence of one event does not affect the probability another event will occur. The average rate (events per time period) is constant. Two events cannot occur at the same time.

### How do you derive Poisson distribution from exponential? ›

Relation between the Poisson and exponential distributions

The Poisson distribution describing this process is therefore P(x) = e−λt(λt)x/x!, from which P (x = 0) = e−λt is the probability of no occurrences in t units of time.

### What are the four properties of Poisson distribution? ›

Properties of Poisson Distribution

The events are independent. The average number of successes in the given period of time alone can occur. No two events can occur at the same time. The Poisson distribution is limited when the number of trials n is indefinitely large.

### What are the main features of Poisson distribution? ›

Characteristics of the Poisson Distribution

As we can see, only one parameter λ is sufficient to define the distribution. ⇒ The mean of X \sim P(\lambda) is equal to λ. ⇒ The variance of X \sim P(\lambda) is also equal to λ. The standard deviation, therefore, is equal to +√λ.

### What is application of Poisson distribution? ›

Companies can utilize the Poisson Distribution to examine how they may be able to take steps to improve their operational efficiency. For instance, an analysis done with the Poisson Distribution might reveal how a company can arrange staffing in order to be able to better handle peak periods for customer service calls.

### Why do we use Poisson distribution? ›

Poisson distributions are used when the variable of interest is a discrete count variable. Many economic and financial data appear as count variables, such as how many times a person becomes unemployed in a given year, thus lending themselves to analysis with a Poisson distribution.

### What is the shape of a Poisson distribution? ›

Unlike a normal distribution, which is always symmetric, the basic shape of a Poisson distribution changes. For example, a Poisson distribution with a low mean is highly skewed, with 0 as the mode. All the data are “pushed” up against 0, with a tail extending to the right.

### What are the assumptions of Poisson distribution? ›

The Poisson distribution is an appropriate model if the following assumptions are true: k is the number of times an event occurs in an interval and k can take values 0, 1, 2, …. The occurrence of one event does not affect the probability that a second event will occur. That is, events occur independently.

### Is Poisson discrete or continuous? ›

The Poisson distribution is a discrete distribution that measures the probability of a given number of events happening in a specified time period.

### What is the real life example of Poisson distribution? ›

the number of Airbus 330 aircraft engine shutdowns per 100,000 flight hours. the number of asthma patient arrivals in a given hour at a walk-in clinic. the number of hungry persons entering McDonald's restaurant per day. the number of work-related accidents over a given production time.

### What is the relationship between Poisson and exponential distribution? ›

Just so, the Poisson distribution deals with the number of occurrences in a fixed period of time, and the exponential distribution deals with the time between occurrences of successive events as time flows by continuously.

### What is lambda in Poisson distribution? ›

In the Poisson distribution formula, lambda (λ) is the mean number of events within a given interval of time or space. For example, λ = 0.748 floods per year.

### Is Poisson a special case of exponential? ›

Poisson is for discrete data and is based on arrivals or occurrence of events. Exponential talks about the time between occurrence of events and is a Continuous Probability Distribution Function. You can look at these two models as they are telling the same story but from different perspectives.

### What are the two main characteristics of a Poisson experiment? ›

Characteristics of a Poisson distribution: The experiment consists of counting the number of events that will occur during a specific interval of time or in a specific distance, area, or volume. The probability that an event occurs in a given time, distance, area, or volume is the same.

### How is Poisson calculated? ›

The Poisson Distribution formula is: P(x; μ) = (e-μ) (μx) / x! Let's say that that x (as in the prime counting function is a very big number, like x = 10100. If you choose a random number that's less than or equal to x, the probability of that number being prime is about 0.43 percent.

### What is the variance of a Poisson distribution? ›

For a Poisson distribution, the variance is given by V(X)=λ=rt V ( X ) = λ = r t where λ is the average number of occurrences of the event in the given time period, r is the average rate of the occurrence of the events, and t is the length of the given time period.

### How do you solve Poisson distribution problems? ›

The formula for Poisson Distribution formula is given below: P ( X = x ) = e − λ λ x x ! x is a Poisson random variable. e is the base of logarithm and e = 2.71828 (approx).

### What is standard deviation of Poisson distribution? ›

= (np)1/2 = µ1/2. The standard deviation is equal to the square-root of the mean. The Poisson distribution is discrete: P(0; µ) = e-µ is the probability of 0 successes, given that the mean number of successes is µ, etc.

### How do you find the probability of a Poisson distribution? ›

Suppose we conduct a Poisson experiment, in which the average number of successes within a given region is μ. Then, the Poisson probability is: P(x; μ) = (e-μ) (μx) / x! where x is the actual number of successes that result from the experiment, and e is approximately equal to 2.71828.

### What is the disadvantages of Poisson distribution? ›

One disadvantage of the Poisson is that it makes strong assumptions regarding the distribution of the underlying data (in particular, that the mean equals the variance). While these assumptions are tenable in some settings, they are less appropriate for alcohol consumption.

### What follows a Poisson distribution? ›

A variable follows a Poisson distribution when the following conditions are true: Data are counts of events. All events are independent. The average rate of occurrence does not change during the period of interest.

### How do you say Poisson? ›

How to Pronounce Poisson? (Distribution, Equation, French)

### Why is Poisson distribution positively skewed? ›

Hence Poisson distribution is always a positively skewed distribution as m>0 as well as leptokurtic. As the value of m increases γ1 decreases and the thus skewness is reduced for increasing values of m. As m⟶∞, γ1 and γ2 tend to zero.

### What is the difference between Binomial and Poisson distribution? ›

Binomial distribution describes the distribution of binary data from a finite sample. Thus it gives the probability of getting r events out of n trials. Poisson distribution describes the distribution of binary data from an infinite sample. Thus it gives the probability of getting r events in a population.

### What is skewness of Poisson distribution? ›

Let X be a discrete random variable with a Poisson distribution with parameter λ. Then the skewness γ1 of X is given by: γ1=1√λ

### How do you read a Poisson distribution table? ›

Poisson Distribution: Using the Probability Tables - YouTube

### What's the difference between Poisson and Gaussian? ›

The Poisson function is defined only for a discrete number of events, and there is zero probability for observing less than zero events. The Gaussian function is continuous and thus takes on all values, including values less than zero as shown for the µ = 4 case.

### Where does a Poisson distribution peak? ›

All Poisson probability functions pλ rise to a peak and then fall again. The peak (a mode) occurs at the greatest integer less than λ, written ⌊λ⌋. When λ is an integer, the peak occurs at the two neighboring values ⌊λ⌋ and ⌊λ⌋+1.

### What is Poisson distribution in machine learning? ›

What is a Poisson Distribution? A Poisson Distribution is a statistical distribution used to express the probability of a given number of events occurring within a fixed interval of time or space. Additionally, the events must occur independently of each other and with a known constant rate.

### How do you derive a binomial distribution? ›

The Binomial Distribution: Mathematically Deriving the Mean ...

### What is a Poisson distribution in statistics? ›

The Poisson distribution is a discrete distribution that measures the probability of a given number of events happening in a specified time period.

### How do you know if a distribution is Poisson? ›

A variable follows a Poisson distribution when the following conditions are true: Data are counts of events. All events are independent. The average rate of occurrence does not change during the period of interest.

### How do you find the probability of a Poisson distribution? ›

Suppose we conduct a Poisson experiment, in which the average number of successes within a given region is μ. Then, the Poisson probability is: P(x; μ) = (e-μ) (μx) / x! where x is the actual number of successes that result from the experiment, and e is approximately equal to 2.71828.

### What is the difference between Poisson and binomial distribution? ›

Binomial distribution describes the distribution of binary data from a finite sample. Thus it gives the probability of getting r events out of n trials. Poisson distribution describes the distribution of binary data from an infinite sample. Thus it gives the probability of getting r events in a population.

### What is the variance of Poisson distribution? ›

The expected value of the Poisson distribution is given as follows: E(x) = μ = d(eλ(t-1))/dt, at t=1. Therefore, the expected value (mean) and the variance of the Poisson distribution is equal to λ.

### What is Poisson distribution find its mean and variance? ›

If \mu is the average number of successes occurring in a given time interval or region in the Poisson distribution. Then the mean and the variance of the Poisson distribution are both equal to \mu.

### Why do we use Poisson distribution? ›

Poisson distributions are used when the variable of interest is a discrete count variable. Many economic and financial data appear as count variables, such as how many times a person becomes unemployed in a given year, thus lending themselves to analysis with a Poisson distribution.

### What are the applications of Poisson distribution? ›

Companies can utilize the Poisson Distribution to examine how they may be able to take steps to improve their operational efficiency. For instance, an analysis done with the Poisson Distribution might reveal how a company can arrange staffing in order to be able to better handle peak periods for customer service calls.

### What are the assumptions of Poisson distribution? ›

The Poisson distribution is an appropriate model if the following assumptions are true: k is the number of times an event occurs in an interval and k can take values 0, 1, 2, …. The occurrence of one event does not affect the probability that a second event will occur. That is, events occur independently.

### What is the meaning of Poisson? ›

noun. : a probability density function that is often used as a mathematical model of the number of outcomes obtained in a suitable interval of time and space, that has its mean equal to its variance, that is used as an approximation to the binomial distribution, and that has the form f(x)=e−μμxx!

### What is true for Poisson distribution? ›

In a Poisson Distribution, the mean and variance are equal. ∴ Mean = Variance.

### What is lambda in Poisson? ›

In the Poisson distribution formula, lambda (λ) is the mean number of events within a given interval of time or space. For example, λ = 0.748 floods per year.

### How do you solve Poisson distribution problems? ›

The formula for Poisson Distribution formula is given below: P ( X = x ) = e − λ λ x x ! x is a Poisson random variable. e is the base of logarithm and e = 2.71828 (approx).

### What is a Poisson distribution PDF? ›

The Poisson distribution is a probability model which can be used to find the probability of a single event occurring a given number of times in an interval of (usually) time.

## Videos

1. Poisson process 1 | Probability and Statistics | Khan Academy
2. Lecture 24: Gamma distribution and Poisson process | Statistics 110
(Harvard University)
3. Poisson Derivation
(MathHolt)
4. [Derivation] Poisson Distribution
(Michael)
5. Developing Binomial Distribution Intuition
(Joel Speranza Math)
6. Derivation of Poisson Distribution
(Maths Online)

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