# 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 is required.
is mean of the distribution.

- Discrete distribution defined at non-negative integer

### Probability

- Cumulative distribution function
- Probability mass function
- How to compute these on Excel.

1 2 3 4 5

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

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

- Standard deviation is given as
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
- How to compute this on Excel

1 2 3 4 A B Data Description 8 Value of parameter nu Formula Description (Result) =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
- How to compute this on Excel

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

## Random Numbers

- How to generate random numbers.

1 2 3 4

A B Data Description 6 Value of parameter nu Formula Description (Result) =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