Poisson Distribution

Abdullah Al Mahmud

Invalid Date

Concept

Used in situation where some events occur at certain place or time.

See applications

Examples

No. of car accidents in a month
No. of sixes in an innings
No. of calls in an hour in a call center
No. of bacteria in a \(mm^2\)
No. of defective bolts in a lot (in statistical quality control)
No. of particles in a radioactive decay
No. of rats in cultivable land
No. of meteorites greater than 1 meter diameter that strike Earth in a year
No. of patients arriving in an emergency room between 10 and 11 pm
No. of laser photons hitting a detector in a particular time interval

Function

\(\displaystyle P(x) = \frac{e^{-\lambda}\lambda^x}{x!}\)

\(e=2.718\) (constant)
x: number of occurrences (success)
\(\lambda\): average no. of events

Poisson Process

A process involving ..

Assumptions

Occurrences are independent
\(p \space \propto\)Interval
another

Properties

Mean and variance are same (\(\lambda\))
mgf: \(e^\lambda e(^t-1)\)
If \(x_1 \sim Poisson(\lambda_1)\) & \(x_2 \sim Poisson(\lambda_1)\), then \((x_1+x_2) \sim Poisson(\lambda_1+\lambda_2)\)
skewness: \(\frac 1 \lambda\) (-ve skew)
kurtosis: platykurtic (\(1+\frac 1 {\sqrt{\lambda}}\))

Theorems

Summation

\(\displaystyle \sum_{i=1}^{\infty} \frac{e^{-\lambda}\lambda^x}{x!}=1\)

Prove

Recurrence

P(x+1)

Poisson to Normal

If \(\lambda\) is very large.

Bionomial to Poisson

Number of trials, n, is very large: \(n \to \infty\)
Probability of success, p, is very low: \(p \to 0\)
Mean, \(np=\lambda\) finite

When to Use

Events that can happen a very large number of time, but happen rarely.
That is, they are used in situations that would be more properly represented by a Binomial distribution with a very large n and small p, especially when the exact values of n and p are unknown.

Difference with Binomial

Binomial counts discrete occurrences among discrete trials (finite attempts)
Poisson counts discrete occurrences among continuous trials (infinite attempts)
In Poisson, success can occur at any point of time (or space)
Accidents in road (anywhere anytime), knots on a rope

Difference with Binomial (Contd.)

Both measure the number of certain random events (or “successes”) within a certain frame - Binomial is based on discrete events, while the Poisson is based on continuous events.
In a Binomial distribution you have a certain number, n, of “attempts,” each of which has probability of success p.
With a Poisson distribution, you essentially have infinite attempts, with infinitesimal chance of success.
Given a Binomial distribution with some n,p, if you let n→∞ and p→0 in such a way that np→λ, then that distribution approaches a Poisson distribution with parameter λ.

Problems

Problem 00

An emergency room receives an average of 3.8 patient arrivals per hour during nighttime (2200 hrs to 0600 hrs).

What is the probability that the number of arrivals is exactly 3?
Find the probability that the number of patient arrivals in an hour is between 2 and 4 (inclusive).
What is the probability that the number of arrivals exceeds 2?

Problem 01

If P(x = 2) = P(x = 3), find

parameters
\(P(x>0)\)
\(P(x\le 2)\)
\(P(x\ge 2)\)

Problem 02

Standard deviation of a Poisson distribution is 4. Find mean and the probabilities in problem 01.

Problem 03

If \(\frac{k\mu^x}{x!}; x = 0, 1, 2, \cdots, \infty,\)

k=?

\(\sum_{x=0}^{\infty}\frac{\mu^x}{x!}=e^{\mu}\)
\(k e^{\mu}=1 \quad\Rightarrow\quad k=e^{-\mu}.\)
\(\boxed{k=e^{-\mu}}\)

Problem 04

If P(a) = P(a+1), what is the value of a?

Because the average event rate is one overflow flood per 100 years, λ = 1

problem 05

The number of machine failures in a factory follows a Poisson distribution with a standard deviation of 2.5.

Find \(P(X \geq 4)\).
If \(P(a) = P(a+2)\), what is the value of \(a\)? Does it hold for Poisson distribution?

Problem 06

The number of device failures within a certain period in a tech industry follows a Poisson distribution with a standard deviation of 2.5.

Find \(P(X \geq 3)\).
What is the probabiility that there no more than 2 device failures within the given period?

Problem 07

Between 1000 hrs and 1700 hrs, the average number of phonce calls per minute received by a power distribution company is 2.5.

Give an example where Poisson distribution is applicable.
Find the relationship between expectation and standard deviation of Poisson distribution.
Find the probability that the number of calls is between 1 and 3 (inclusive).
What is the probability that the number of calls received is greater than the average?

Problem 08

A can manufacturer observes that 0.1% of the produced cans are faulty. The cans are packaged in carton boxes, with 500 cans in each box. A wholeseller purchases 100 boxes from the manufacturer.

When can the Poisson distribution be approximated by the Binomial distribution?
Find the probability of exactly one defective can (in a box), using the Poisson approximation.
Find the expected number of boxes with no defective cans.

Problem 09

The frequency distribution of defective items in packets of key rings is given below.

Number of defective items	0	1	2	3	4	5
Number of packets	76	74	29	17	3	1

From the given data, find the mean and variance.
Fit a Poisson distribution and find the expected frequencies. Comment on the fit.

Problem 10

The number of customers coming at a supershop follows a Poisson distribution with mean 3.

Determine the probability that in a particular minute, at least 1 customer arrives.
If \(P(X=a) = P(X=b)\), find the values of \(a\) and \(b\). What pattern do you observe?

Question Bank

Practice Online

Practice Poisson Distribution MCQs interactively (with instant feedback): Stat & Prob Quiz

Pick Statistics II as subject and Poisson Distribution as chapter/topic.

Question Bank Archive

MCQ Bank — Poisson Distribution section starts at the relevant chapter.
CQ Bank — creative questions for Poisson Distribution.