Poisson Distribution

Where do you meet this distribution?

• Accident : The number of soldiers killed by horse-kicks each year in each corps
• Queuing Theory : Number of phone calls per minute, number of access to web sever per minute
• Biology : Number of mutations
• Nuclear physics : the nuclear decay of atoms
• Risk management -- Operational risk

Shape of Distribution

Basic Properties

• A parameter $\nu$ is required.

$$\nu>0$$

$\nu$ is mean of the distribution.

• Discrete distribution defined at non-negative integer $x={0,1,2,\cdots}$

Probability

• Cumulative distribution function

  $$F(x)=\text{e}^{-\nu}\sum_{i=0}^{x}\frac{\nu^i}{i !}$$

• Probability mass function

  $$f(x)=\frac{\nu^x\text{e}^{-\nu}}{x !}$$

• How to compute these on Excel.

	A	B
1	Data	Description
2	3	Value for which you want the distribution
3	5	Value of parameter nu
4	Formula	Description (Result)
5	=NTPOISSONDIST(A2,A3,TRUE)	Cumulative distribution function for the terms above
6	=NTPOISSONDIST(A2,A3,FALSE)	Probability mass function for the terms above

• Function reference : NTPOISSONDIST

Characteristics

Mean -- Where is the "center" of the distribution? (Definition)

• Mean is given as $\nu$

Standard Deviation -- How wide does the distribution spread? (Definition)

• Standard deviation is given as $\nu$

  Standard Deviation is a positive square root of Variance.

Skewness -- Which side is the distribution distorted into? (Definition)

• Skewness of the distribution is given as

  $$\frac{1}{\sqrt{\nu}}$$

• How to compute this on Excel

	A	B
1	Data	Description
2	8	Value of parameter nu
3	Formula	Description (Result)
4	=NTPOISSONSKEW(A2)	Mean of the distribution for the terms above

• Function reference : NTPOISSONSKEW

Kurtosis -- Sharp or Dull, consequently Fat Tail or Thin Tail (Definition)

• Kurtosis of the distribution is given as

  $$\frac{1}{\nu}$$

• How to compute this on Excel

	A	B
1	Data	Description
2	8	Value of parameter nu
3	Formula	Description (Result)
4	=NTPOISSONKURT(A2)	Mean of the distribution for the terms above

• Function reference : NTPOISSONKURT

Random Numbers

• How to generate random numbers.

	A	B
1	Data	Description
2	6	Value of parameter nu
3	Formula	Description (Result)
4	=NTRANDPOISSON(100,A2,0)	100 Poisson deviates based on Mersenne-Twister algorithm for which the parameters above

Note The formula in the example must be entered as an array formula. After copying the example to a blank worksheet, select the range A4:A103 starting with the formula cell. Press F2, and then press CTRL+SHIFT+ENTER.

NtRand Functions

• If you already have parameters of the distribution
  • Generating random numbers based on Mersenne Twister algorithm: NTRANDPOISSON
  • Computing probability : NTPOISSONDIST
  • Computing mean : NTPOISSONMEAN
  • Computing standard deviation : NTPOISSONSTDEV
  • Computing skewness : NTPOISSONSKEW
  • Computing kurtosis : NTPOISSONKURT
  • Computing moments above at once : NTPOISSONMOM

Reference

• Wolfram Mathworld -- Poisson Distribution
• Wikipedia -- Poisson distribution
• Statistics Online Computational Resource