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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]</div> These parameters are minimum and maximum value of variable respectively </li> <li>Discrete distribution defined at [latex]x=\{a, a+1, \cdots, b\}

Probability

  • Cumulative distribution function
    F(x)=\begin{cases}0\;&(x<a)\\\frac{x-a+1}{b-a+1}\;&(x=\{a,a+1,\cdots,b\})\\1\;&(x>b)\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

Reference

 

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