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Chi Distribution

Shape of Distribution​

Basic Properties​

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

Probability​

  • Probability density function

    f(x)=12N2−1Γ(N2)exp⁡(−x22)xN−1f(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 Γ(⋅)\Gamma(\cdot) is gamma function.

  • Cumulative distribution function

    F(x)=Γx22(N2)Γ(N2)F(x)=\frac{\Gamma_{\frac{x^2}{2}}\left(\frac{N}{2}\right)}{\Gamma\left(\frac{N}{2}\right)}

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

  • How to compute these on Excel.

AB
1DataDescription
25Value for which you want the distribution
39Value of parameter N
4FormulaDescription (Result)
5=NTCHIDIST(A2,A3,TRUE)Cumulative distribution function for the terms above
6=NTCHIDIST(A2,A3,FALSE)Probability density function for the terms above

Characteristics​

Mean -- Where is the "center" of the distribution? (Definition)​

  • Mean of the distribution is given as

    2Γ(N+12)Γ(N2)\sqrt{2}\frac{\Gamma\left(\frac{N+1}{2}\right)}{\Gamma\left(\frac{N}{2}\right)}

    , where Γ(⋅)\Gamma(\cdot) is gamma function.

  • How to compute this on Excel

AB
1DataDescription
28Value of parameter N
3FormulaDescription (Result)
4=NTCHIMEAN(A2)Mean of the distribution for the terms above

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

  • Variance of the distribution is given as

    N−2[Γ(N+12)Γ(N2)]2N-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

AB
1DataDescription
28Value of parameter N
3FormulaDescription (Result)
4=NTCHISTDEV(A2)Standard deviation of the distribution for the terms above

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

  • Skewness of the distribution is given as

    μ(3)−3μ(2)μ(1)+2μ3(1)σ3\frac{\mu(3)-3\mu(2)\mu(1)+2\mu^3(1)}{\sigma^3} μ(r)=2r2Γ(N+r2)Γ(N2)\mu(r)=\frac{2^{\frac{r}{2}}\Gamma\left(\frac{N+r}{2}\right)}{\Gamma\left(\frac{N}{2}\right)}

    , where σ\sigma is standard deviation and Γ(⋅)\Gamma(\cdot) is gamma function.

  • How to compute this on Excel

AB
1DataDescription
28Value of parameter N
3FormulaDescription (Result)
4=NTCHISKEW(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

    μ(4)−4μ(3)μ(1)+6μ(2)μ2(1)−3μ4(1)σ4−3\frac{\mu(4)-4\mu(3)\mu(1)+6\mu(2)\mu^2(1)-3\mu^4(1)}{\sigma^4}-3 μ(r)=2r2Γ(N+r2)Γ(N2)\mu(r)=\frac{2^{\frac{r}{2}}\Gamma\left(\frac{N+r}{2}\right)}{\Gamma\left(\frac{N}{2}\right)}

    , where σ\sigma is standard deviation and Γ(⋅)\Gamma(\cdot) is gamma function.

  • This distribution can be leptokurtic or platykurtic.

  • How to compute this on Excel

AB
1DataDescription
28Value of parameter N
3FormulaDescription (Result)
4=NTCHIKURT(A2)Kurtosis of the distribution for the terms above

Random Numbers​

  • How to generate random numbers on Excel.
AB
1DataDescription
28Value of parameter N
3FormulaDescription (Result)
4=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.

NtRand Functions​

  • If you already have parameters of the distribution

Reference​