T distribution

Shape of Distribution

Basic Properties

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

Probability

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

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

• Cumulative distribution function
$F(x)=\frac{1}{2}-\frac{1}{2}\left[1-I_{\gamma}\left(\frac{1}{2},\frac{N}{2}\right)\right]\text{sign}(x)$

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

• How to compute these on Excel.

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

Characteristics

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

• Mean of the distribution is defined for $N>1$ and is always 0.
• How to compute this on Excel

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

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

• Variance of the distribution is given as
$\frac{N}{N-2}\quad (N>2)$

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)
=NTTSTDEV(A2) Standard deviation of the distribution for the terms above
• Function reference : NTTSTDEV

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

• Skewness of the distribution is defined for $N>3$ and is always 0.
$\sqrt{\frac{8}{N}}$
• How to compute this on Excel

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

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

• Kurtosis of the distribution is given as
$\frac{6}{N-4}\;(N>4)$
• 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)
=NTTKURT(A2) Kurtosis of the distribution for the terms above
• Function reference : NTTKURT

Random Numbers

• How to generate random numbers on Excel.

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Data Description
9 Value of parameter N
Formula Description (Result)
=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.

• Function reference : NTRANDT

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