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Poisson Distribution

Abdullah Al Mahmud

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 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=?

Problem 04

Overflow floods occur once every 100 years on average. Calculate the probability of k = 0, 1, 2, 3, 4, 5, or 6 overflow floods in a 100-year interval, assuming the Poisson model is appropriate.

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

problem 05

Ugarte and colleagues report that the average number of goals in a World Cup soccer match is approximately 2.5 and the Poisson model is appropriate. Estimate probability of k goals and then k = 0,1,2,3..

Poisson Distribution Abdullah Al Mahmud