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T distribution

Shape of Distribution​

Basic Properties​

  • One parameter NN is required (Positive integer)
  • Continuous distribution defined on on entire range
  • This distribution is symmetric.

Probability​

  • Probability density function

    f(x)=Γ(N+12)πN(1+x2N)N+1Γ(N2)f(x)=\frac{\Gamma\left(\frac{N+1}{2}\right)}{\sqrt{\pi N\left(1+\frac{x^2}{N}\right)^{N+1}}\Gamma\left(\frac{N}{2}\right)}

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

  • Cumulative distribution function

    F(x)=12−12[1−Iγ(12,N2)]sign(x)F(x)=\frac{1}{2}-\frac{1}{2}\left[1-I_{\gamma}\left(\frac{1}{2},\frac{N}{2}\right)\right]\text{sign}(x)

    , where γ=NN+x2\gamma=\frac{N}{N+x^2} and Ix(⋅,⋅)I_{x}(\cdot,\cdot) is regularized incomplete beta function.

  • How to compute these on Excel.

AB
1DataDescription
25Value for which you want the distribution
38Value of parameter N
4FormulaDescription (Result)
5=NTTDIST(A2,A3,TRUE)Cumulative distribution function for the terms above
6=NTTDIST(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 defined for N>1N>1 and is always 0.
  • How to compute this on Excel
AB
1DataDescription
28Value of parameter N
3FormulaDescription (Result)
4=NTTMEAN(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

    NN−2(N>2)\frac{N}{N-2}\quad (N>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=NTTSTDEV(A2)Standard deviation of the distribution for the terms above

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

  • Skewness of the distribution is defined for $N>3$$ and is always 0.

    8N\sqrt{\frac{8}{N}}

  • How to compute this on Excel

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

    6N−4;(N>4)\frac{6}{N-4};(N>4)

  • This distribution can be leptokurtic or platykurtic.

  • How to compute this on Excel

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

Random Numbers​

  • How to generate random numbers on Excel.
AB
1DataDescription
29Value of parameter N
3FormulaDescription (Result)
4=NTRANDT(100,A2,0)100 chi square 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
    • Generating random numbers based on Mersenne Twister algorithm: NTRANDT
    • Computing probability : NTTDIST
    • Computing mean : NTTMEAN
    • Computing standard deviation : NTTSTDEV
    • Computing skewness : NTTSKEW
    • Computing kurtosis : NTTKURT
    • Computing moments above at once : NTTMOM

Reference​