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Statistics/Distributions/Student-t

Student-t DistributionEdit

Student’s t
Probability density function
 
Cumulative distribution function
 
Parameters ν > 0 degrees of freedom (real)
Support x ∈ (−∞; +∞)
PDF  
CDF  
where 2F1 is the hypergeometric function
Mean 0 for ν > 1, otherwise undefined
Median 0
Mode 0
Variance   for ν > 2, ∞ for 1 < ν ≤ 2, otherwise undefined
Skewness 0 for ν > 3, otherwise undefined
Ex. kurtosis   for ν > 4, ∞ for 2 < ν ≤ 4, otherwise undefined
Entropy ...
MGF undefined
CF   for ν > 0

Student t-distribution (or just t-distribution for short) is derived from the chi-square and normal distributions. We divide the standard normally distributed value of one variable over the root of a chi-square value over its r degrees of freedom. Mathematically, this appears as:

 

where   and  .

External linksEdit

  • Hurst, Simon, The Characteristic Function of the Student-t Distribution, Financial Mathematics Research Report No. FMRR006-95, Statistics Research Report No. SRR044-95