Poisson Distribution

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

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..