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Beta distribution

Where do you meet this distribution?

Shape of Distribution

Basic Properties

  • Two parameters are required (How can you get these?)
  • Continuous distribution defined on bounded range
  • This distribution can be symmetric or asymmetric.

Probability

Quantile

  • Inverse function of cumulative distribution function cannot be expressed in closed form.
  • BETAINV is an excel function.
  • How to compute this on Excel.

     
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    A B
    Data Description
    0.7 Probability associated with the distribution
    1.7 Value of parameter Alpha
    0.9 Value of parameter Beta
    Formula Description (Result)
    =BETAINV(A2,A3,A4) Inverse of the cumulative distribution function for the terms above

Characteristics

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

  • Mean of the distribution is given as
  • How to compute this on Excel

     
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    A B
    Data Description
    8 Value of parameter Alpha
    2 Value of parameter Beta
    Formula Description (Result)
    =NTBETAMEAN(A2,A3) Mean of the distribution for the terms above
  • Function reference : NTBETAMEAN

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

  • Variance of the distribution is given as

    Standard Deviation is a positive square root of Variance.

  • How to compute this on Excel

     
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    A B
    Data Description
    8 Value of parameter Alpha
    2 Value of parameter Beta
    Formula Description (Result)
    =NTBETASTDEV(A2,A3) Standard deviation of the distribution for the terms above
  • Function reference : NTBETASTDEV

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

  • Skewness of the distribution is given as
  • How to compute this on Excel

     
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    A B
    Data Description
    8 Value of parameter Alpha
    2 Value of parameter Beta
    Formula Description (Result)
    =NTBETASKEW(A2,A3) Skewness of the distribution for the terms above
  • Function reference : NTBETASKEW

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

  • Kurtosis of the distribution is given as
  • This distribution can be leptokurtic or platykurtic.
  • How to compute this on Excel

     
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    A B
    Data Description
    8 Value of parameter Alpha
    2 Value of parameter Beta
    Formula Description (Result)
    =NTBETAKURT(A2,A3) Kurtosis of the distribution for the terms above
  • Function reference : NTBETAKURT

Random Numbers

  • The algorithm to generated random numbers is shown in:

    R. C. H. Cheng, “Generating beta variates with nonintegral shape parameters”, Communication of the ACM, 21(1978), pp 317-322

  • How to generate random numbers on Excel.
     
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    A B
    Data Description
    0.5 Value of parameter Alpha
    0.5 Value of parameter Beta
    Formula Description (Result)
    =NTRANDBETA(100,A2,A3,0) 100 beta 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.

  • Function reference : NTRANDBETA

NtRand Functions

  • If you already have parameters of the distribution
  • If you know mean and standard deviation of the distribution
    • Estimating parameters of the distribution:NTBETAPARAM

Reference

 

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