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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

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)

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

Reference

 

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