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# Gamma 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 semi-bounded range $x \geq 0$
• This distribution is asymmetric.

### Probability

• Probability density function
$f(x)=\frac{1}{\beta^{\alpha}\Gamma(\alpha)}\exp\left(-\frac{x}{\beta}\right)x^{\alpha-1}$

, where $\Gamma(\cdot)$ is gamma function.

• Cumulative distribution function
$F(x)=\frac{\Gamma_{\frac{x}{\beta}}(\alpha)}{\Gamma(\alpha)}$

, where $\Gamma_{x}(\cdot)$ is incomplete gamma function.

• How to compute these on Excel.

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

### Quantile

• Inverse function of cumulative distribution function cannot be expressed in closed form.
• GAMMAINV 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
4 Value of parameter Alpha
2.3 Value of parameter Beta
Formula Description (Result)
=GAMMAINV(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
$\alpha\beta$
• How to compute this on Excel

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

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

• Variance of the distribution is given as
$\alpha\beta^2$

Standard Deviation is a positive square root of Variance.

• How to compute this on Excel

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

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

• Skewness of the distribution is given as
$\frac{2}{\sqrt{\alpha}}$
• How to compute this on Excel

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A B
Data Description
4 Value of parameter Alpha
Formula Description (Result)
=NTGAMMASKEW(A2) 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
$\frac{6}{\alpha}$
• This distribution can be leptokurtic or platykurtic.
• How to compute this on Excel

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A B
Data Description
4 Value of parameter Alpha
Formula Description (Result)
=NTGAMMAKURT(A2) Kurtosis of the distribution for the terms above
• Function reference : NTGAMMAKURT

## Random Numbers

• How to generate random numbers on Excel.

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A B
Data Description
4 Value of parameter Alpha
2.3 Value of parameter Beta
Formula Description (Result)
=NTRANDGAMMA(100,A2,A3,0) 100 gamma 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 : NTRANDGAMMA

## NtRand Functions

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