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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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A	B
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
- Generating random numbers based on Mersenne Twister algorithm: NTRANDCHI
- Computing probability : NTCHIDIST
- Computing mean : NTCHIMEAN
- Computing standard deviation : NTCHISTDEV
- Computing skewness : NTCHISKEW
- Computing kurtosis : NTCHIKURT
- Computing moments above at once : NTCHIMOM

Reference

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Chi distribution

Shape of Distribution

Basic Properties

Probability

Characteristics

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

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

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

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

Random Numbers

NtRand Functions

Reference

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Willing to generate Random Numbers in Excel?

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