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# Uniform distribution (Discrete)

## Where do you meet this distribution?

• Gambling, Game : Dice, Roulette
• n-dim. random walk
• Fisher-Yates shuffle algorithm

## Shape of Distribution

### Basic Properties

• Two integer parameters $a,b$ is required.
$a < b[/latex] These parameters are minimum and maximum value of variable respectively
• Discrete distribution defined at [latex]x=\{a, a+1, \cdots, b\}$

### Probability

• Cumulative distribution function
$F(x)=\begin{cases}0\;&(xb)\end{cases}$
• Probability mass function
$f(x)=\begin{cases}\frac{1}{b-a+1}\;&(x=\{a,a+1,\cdots,b\})\\0\;&(\text{otherwise})\end{cases}$
• How to compute these on Excel.

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A B
Data Description
3 Value for which you want the distribution
1 Value of parameter A
6 Value of parameter B
Formula Description (Result)
=IF(A2<A3,0,IF(A2<=A4, (A2-A3+1)/(A4-A3+1),1)) Cumulative distribution function for the terms above
=IF(AND(A3<=A2,A2<=A4),1/(A4-A3+1), 0) Probability mass function for the terms above

## Characteristics

### Mean – Where is the “center” of the distribution? (Definition)

• Mean of the distribution is given as
$\frac{a+b}{2}$
• How to compute this on Excel

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A B
Data Description
1 Value of parameter A
6 Value of parameter B
Formula Description (Result)
=(A2+A3)/2 Mean of the distribution for the terms above

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

• Variance of the distribution is given as
$\frac{(b-a+1)^2-1}{12}$

Standard Deviation is a positive square root of Variance

• How to compute this on Excel

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A B
Data Description
1 Value of parameter A
6 Value of parameter B
Formula Description (Result)
=SQRT(((A3-A2+1)^2-1)/12) Standard deviation of the distribution for the terms above

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

• Skewness is $0$ .

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

• Kurtosis of the distribution is given as
$-\frac{6((b-a+1)^2+1}{5((b-a+1)^2-1}$
• How to compute this on Excel

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A B
Data Description
1 Value of parameter A
6 Value of parameter B
Formula Description (Result)
=-6*((A3-A2+1)^2+1)/(5*((A3-A2+1)^2-1)) Kurtosis of the distribution for the terms above

## Random Numbers

• How to generate random numbers on Excel.

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A B
Data Description
1 Value of parameter A
6 Value of parameter B
Formula Description (Result)
=INT((A3-A2+1)*NTRAND(100))+A2 100 uniform 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 A5:A104 starting with the formula cell. Press F2, and then press CTRL+SHIFT+ENTER.

## NtRand Functions

Not supported yet