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

Events that can happen a very large number of time, but happen rarely.

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

Problems

Problem 01

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

  1. parameters
  2. \(P(x>0)\)
  3. \(P(x\le 2)\)
  4. \(P(x\ge 2)\)

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..