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# Chi distribution

## Shape of Distribution

### Basic Properties

• One parameter $N$ is required (Positive integer)
• Continuous distribution defined on semi-bounded range $x \geq 0$
• This distribution is asymmetric.

### Probability

• Probability density function
$f(x)=\frac{1}{2^{\frac{N}{2}-1}\Gamma\left(\frac{N}{2}\right)}\exp\left(-\frac{x^2}{2}\right)x^{N-1}$

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

• Cumulative distribution function
$F(x)=\frac{\Gamma_{\frac{x^2}{2}}\left(\frac{N}{2}\right)}{\Gamma\left(\frac{N}{2}\right)}$

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

• How to compute these on Excel.

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Data Description
5 Value for which you want the distribution
9 Value of parameter N
Formula Description (Result)
=NTCHIDIST(A2,A3,TRUE) Cumulative distribution function for the terms above
=NTCHIDIST(A2,A3,FALSE) Probability density function for the terms above
• Function reference : NTCHIDIST

## Characteristics

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

• Mean of the distribution is given as
$\sqrt{2}\frac{\Gamma\left(\frac{N+1}{2}\right)}{\Gamma\left(\frac{N}{2}\right)}$

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

• How to compute this on Excel

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Data Description
8 Value of parameter N
Formula Description (Result)
=NTCHIMEAN(A2) Mean of the distribution for the terms above
• Function reference : NTCHIMEAN

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

• Variance of the distribution is given as
$N-2\left[\frac{\Gamma\left(\frac{N+1}{2}\right)}{\Gamma\left(\frac{N}{2}\right)}\right]^2$

Standard Deviation is a positive square root of Variance.

• How to compute this on Excel

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Data Description
8 Value of parameter N
Formula Description (Result)
=NTCHISTDEV(A2) Standard deviation of the distribution for the terms above
• Function reference : NTCHISTDEV

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

• Skewness of the distribution is given as
$\frac{\mu(3)-3\mu(2)\mu(1)+2\mu^3(1)}{\sigma^3}$
$\mu(r)=\frac{2^{\frac{r}{2}}\Gamma\left(\frac{N+r}{2}\right)}{\Gamma\left(\frac{N}{2}\right)}$

, where $\sigma$ is standard deviation and $\Gamma(\cdot)$ is gamma function.

• How to compute this on Excel

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Data Description
8 Value of parameter N
Formula Description (Result)
=NTCHISKEW(A2) Skewness of the distribution for the terms above
• Function reference : NTCHISKEW

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

• Kurtosis of the distribution is given as
$\frac{\mu(4)-4\mu(3)\mu(1)+6\mu(2)\mu^2(1)-3\mu^4(1)}{\sigma^4}-3$
$\mu(r)=\frac{2^{\frac{r}{2}}\Gamma\left(\frac{N+r}{2}\right)}{\Gamma\left(\frac{N}{2}\right)}$

, where $\sigma$ is standard deviation and $\Gamma(\cdot)$ is gamma function.

• This distribution can be leptokurtic or platykurtic.
• How to compute this on Excel

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Data Description
8 Value of parameter N
Formula Description (Result)
=NTCHIKURT(A2) Kurtosis of the distribution for the terms above
• Function reference : NTCHIKURT

## Random Numbers

• How to generate random numbers on Excel.

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Data Description
8 Value of parameter N
Formula Description (Result)
=NTRANDCHI(100,A2,0) 100 chi 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 : NTRANDCHI

## NtRand Functions

• If you already have parameters of the distribution