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Beta Distribution

Where do you meet this distribution?​

Shape of Distribution​

Basic Properties​

  • Two parameters α,β\alpha, \beta are required (How can you get these?)

    α>0,β>0\alpha>0,\beta>0

  • Continuous distribution defined on bounded range 0≤x≤10\leq x \leq 1

  • This distribution can be symmetric or asymmetric.

Probability​

AB
1DataDescription
20.5Value for which you want the distribution
38Value of parameter Alpha
42Value of parameter Beta
5FormulaDescription (Result)
6=NTBETADIST(A2,A3,A4,TRUE)Cumulative distribution function for the terms above
7=NTBETADIST(A2,A3,A4,FALSE)Probability density function for the terms above

Beta distribution

Quantile​

AB
1DataDescription
20.7Probability associated with the distribution
31.7Value of parameter Alpha
40.9Value of parameter Beta
5FormulaDescription (Result)
6=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

    αα+β\frac{\alpha}{\alpha+\beta}

  • How to compute this on Excel

AB
1DataDescription
28Value of parameter Alpha
32Value of parameter Beta
4FormulaDescription (Result)
5=NTBETAMEAN(A2,A3)Mean of the distribution for the terms above

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

  • Variance of the distribution is given as

    αβ(α+β)2(α+β+1)\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}

    Standard Deviation is a positive square root of Variance.

  • How to compute this on Excel

AB
1DataDescription
28Value of parameter Alpha
32Value of parameter Beta
4FormulaDescription (Result)
5=NTBETASTDEV(A2,A3)Standard deviation of the distribution for the terms above

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

  • Skewness of the distribution is given as

    2(β−α)α+β+1(α+β+2)αβ\frac{2(\beta-\alpha)\sqrt{\alpha+\beta+1}}{(\alpha+\beta+2)\sqrt{\alpha\beta}}

  • How to compute this on Excel

AB
1DataDescription
28Value of parameter Alpha
32Value of parameter Beta
4FormulaDescription (Result)
5=NTBETASKEW(A2,A3)Skewness of the distribution for the terms above

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

  • Kurtosis of the distribution is given as

    6α3−α2(2β−1)+β2(β+1)−2αβ(β+2)αβ(α+β+2)(α+β+3)6\frac{\alpha^3-\alpha^2(2\beta-1)+\beta^2(\beta+1)-2\alpha\beta(\beta+2)}{\alpha\beta(\alpha+\beta+2)(\alpha+\beta+3)}

  • This distribution can be leptokurtic or platykurtic.

  • How to compute this on Excel

AB
1DataDescription
28Value of parameter Alpha
32Value of parameter Beta
4FormulaDescription (Result)
5=NTBETAKURT(A2,A3)Kurtosis of the distribution for the terms above

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.

AB
1DataDescription
20.5Value of parameter Alpha
30.5Value of parameter Beta
4FormulaDescription (Result)
5=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.

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​