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Student's t-distribution

In probability theory and statistics, Student's t distribution (or simply the t distribution) t ν {\displaystyle t_{\nu }} is a continuous probability distribution that generalizes the standard normal distribution. Like the latter, it is symmetric around zero and bell-shaped.

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CDF
1 2 + x Γ ( ν + 1 2 ) π ν Γ ( ν 2 ) × 2 F 1 ( 1 2 , ν + 1 2 ; 3 2 ; − x 2 ν ) , {\displaystyle {\begin{aligned}&{\frac {1}{2}}+x{\frac {\Gamma {\left({\frac {\nu +1}{2}}\right)}…
CF
( ν | t | ) ν / 2 K ν / 2 ( ν | t | ) Γ ( ν / 2 ) 2 ν / 2 − 1 {\displaystyle {\frac {{\big (}{\sqrt {\nu }}\,|t|{\big )}^{\nu /2}\,K_{\nu /2}{\big (}{\sqrt {\nu }}\,|t|{\big )}}…
Entropy
ν + 1 2 [ ψ ( ν + 1 2 ) − ψ ( ν 2 ) ] + ln ⁡ [ ν B ( ν 2 , 1 2 ) ] (nats) , {\displaystyle {\begin{aligned}&{\frac {\nu +1}{2}}\left[\psi {\left({\frac {\nu +1}{2}}\right)}-\psi…
Excess kurtosis
6 ν − 4 {\displaystyle {\frac {6}{\nu -4}}} for ν > 4 , {\displaystyle \nu >4,} ∞ {\displaystyle \infty } for 2 < ν ≤ 4 , {\displaystyle 2<\nu \leq 4,} otherwise undefined
Mean
0 {\displaystyle 0} for ν > 1 , {\displaystyle \nu >1,} otherwise undefined
Median
0 {\displaystyle 0}

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Student's t-distribution

Nodes109
Edges108
Triples56
Avg. degree1.98
Density0.018349
Components1

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Student's t-distribution

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related to history · 22
Student's t-distribution → Another, As, Biometrika, Dublin, English-language, Eponymy, Gosset's, Guinness, Guinness Brewery, Helmert, In, Ireland, IV, Karl Pearson's, Lüroth, One, Pearson, Stigler's Law, Student, Student's
related to External links · 18
Student's t-distribution → Archived, Distribution, Earliest Known Uses, EMS Press, Encyclopedia, Estimation, First Students, Mathematics, PDF, Probability, Remarks, Rouaud, Some, Statistics, Student, Student's, Wayback Machine, Words
related to Special cases · 2
Student's t-distribution → Certain, Student's
CDF · 1
Student's t-distribution → 1 2 + x Γ ( ν + 1 2 ) π ν Γ ( ν 2 ) × 2 F 1 ( 1 2 , ν + 1 2 ; 3 2 ; − x 2 ν ) , {\displaystyle {\begin{aligned}&{\frac {1}{2}}+x{\frac {\Gamma {\left({\frac {\nu +1}{2}}\right)}…
CF · 1
Student's t-distribution → ( ν | t | ) ν / 2 K ν / 2 ( ν | t | ) Γ ( ν / 2 ) 2 ν / 2 − 1 {\displaystyle {\frac {{\big (}{\sqrt {\nu }}\,|t|{\big )}^{\nu /2}\,K_{\nu /2}{\big (}{\sqrt {\nu }}\,|t|{\big )}}…
Entropy · 1
Student's t-distribution → ν + 1 2 [ ψ ( ν + 1 2 ) − ψ ( ν 2 ) ] + ln ⁡ [ ν B ( ν 2 , 1 2 ) ] (nats) , {\displaystyle {\begin{aligned}&{\frac {\nu +1}{2}}\left[\psi {\left({\frac {\nu +1}{2}}\right)}-\psi…
Excess kurtosis · 1
Student's t-distribution → 6 ν − 4 {\displaystyle {\frac {6}{\nu -4}}} for ν 4 , {\displaystyle \nu 4,} ∞ {\displaystyle \infty } for 2 ν ≤ 4 , {\displaystyle 2\nu \leq 4,} otherwise undefined
Mean · 1
Student's t-distribution → 0 {\displaystyle 0} for ν 1 , {\displaystyle \nu 1,} otherwise undefined
Median · 1
Student's t-distribution → 0 {\displaystyle 0}
MGF · 1
Student's t-distribution → undefined

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distribution displaystyle nu student's frac mean variance normal left right probability sqrt mu freedom function degrees value confidence data gamma

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SubjectPredicateObjectConfidenceSrc
Student's t-distributionCDF1 2 + x Γ ( ν + 1 2 ) π ν Γ ( ν 2 ) × 2 F 1 ( 1 2 , ν + 1 2 ; 3 2 ; − x 2 ν ) , {\displaystyle {\begin{aligned}&{\frac {1}{2}}+x{\frac {\Gamma {\left({\frac {\nu +1}{2}}\right)}…1.00infobox
Student's t-distributionCF( ν | t | ) ν / 2 K ν / 2 ( ν | t | ) Γ ( ν / 2 ) 2 ν / 2 − 1 {\displaystyle {\frac {{\big (}{\sqrt {\nu }}\,|t|{\big )}^{\nu /2}\,K_{\nu /2}{\big (}{\sqrt {\nu }}\,|t|{\big )}}…1.00infobox
Student's t-distributionEntropyν + 1 2 [ ψ ( ν + 1 2 ) − ψ ( ν 2 ) ] + ln ⁡ [ ν B ( ν 2 , 1 2 ) ] (nats) , {\displaystyle {\begin{aligned}&{\frac {\nu +1}{2}}\left[\psi {\left({\frac {\nu +1}{2}}\right)}-\psi…1.00infobox
Student's t-distributionExcess kurtosis6 ν − 4 {\displaystyle {\frac {6}{\nu -4}}} for ν > 4 , {\displaystyle \nu >4,} ∞ {\displaystyle \infty } for 2 < ν ≤ 4 , {\displaystyle 2<\nu \leq 4,} otherwise undefined1.00infobox
Student's t-distributionMean0 {\displaystyle 0} for ν > 1 , {\displaystyle \nu >1,} otherwise undefined1.00infobox
Student's t-distributionMedian0 {\displaystyle 0}1.00infobox
Student's t-distributionMGFundefined1.00infobox
Student's t-distributionMode0 {\displaystyle 0}1.00infobox
Student's t-distributionParametersν > 0 {\displaystyle \nu >0} degrees of freedom (real, nearly always a positive integer)1.00infobox
Student's t-distributionPDFΓ ( ν + 1 2 ) π ν Γ ( ν 2 ) ( 1 + x 2 ν ) − ν + 1 2 {\displaystyle {\frac {\Gamma {\left({\frac {\nu +1}{2}}\right)}}{{\sqrt {\pi \nu }}\,\Gamma {\left({\frac {\nu }{2}}\right)}…1.00infobox
Student's t-distributionSkewness0 {\displaystyle 0} for ν > 3 , {\displaystyle \ \nu >3\ ,} otherwise undefined1.00infobox
Student's t-distributionSupportx ∈ ( − ∞ , ∞ ) {\displaystyle x\in (-\infty ,\infty )}1.00infobox
Student's t-distributionVarianceν ν − 2 {\displaystyle {\frac {\nu }{\nu -2}}} for ν > 2 , {\displaystyle \nu >2,} ∞ {\displaystyle \infty } for 1 < ν ≤ 2 , {\displaystyle 1<\nu \leq 2,} otherwise undefined1.00infobox
Student's t-distributionis aexample of Stigler's Law of Eponymy0.90text

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