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# Poisson distribution

## 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.

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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 $\nu$

### 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

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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
$\frac{1}{\nu}$
• How to compute this on Excel

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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.

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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.