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# Beta distribution

## Shape of Distribution

### Basic Properties

• Two parameters $\alpha, \beta$ are required (How can you get these?)
$\alpha>0,\beta>0$
• Continuous distribution defined on bounded range $0\leq x \leq 1$
• This distribution can be symmetric or asymmetric.

### Probability

• Cumulative distribution function
$F(x)=I_x(\alpha,\beta)$

where $I_x(\cdot,\cdot)$ is regularized incomplete beta function.

• Probability density function
$f(x)=\frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}$

where $B(\cdot,\cdot)$ is beta function.

• When the 4th. argument of NTBETADIST=TRUE, this function returns same result as Excel function “BETADIST” does.
• How to compute these on Excel.

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A B
Data Description
0.5 Value for which you want the distribution
8 Value of parameter Alpha
2 Value of parameter Beta
Formula Description (Result)
=NTBETADIST(A2,A3,A4,TRUE) Cumulative distribution function for the terms above
=NTBETADIST(A2,A3,A4,FALSE) Probability density function for the terms above

### 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
$\frac{\alpha}{\alpha+\beta}$
• 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
$\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}$

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
$\frac{2(\beta-\alpha)\sqrt{\alpha+\beta+1}}{(\alpha+\beta+2)\sqrt{\alpha\beta}}$
• 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

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

• Kurtosis of the distribution is given as
$6\frac{\alpha^3-\alpha^2(2\beta-1)+\beta^2(\beta+1)-2\alpha\beta(\beta+2)}{\alpha\beta(\alpha+\beta+2)(\alpha+\beta+3)}$
• This distribution can be leptokurtic or platykurtic.
• How to compute this on Excel

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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