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Student's t-Distribution

NIST/SEMATECH Section 1.3.6.6.4 Student's t-Distribution

PDF

t(df=5) PDF -4 -2 0 2 4 x 0 0.1 0.2 0.3 0.4 f(x)

CDF

t(df=5) CDF -4 -2 0 2 4 x 0 0.2 0.4 0.6 0.8 1 F(x)

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CDF

Formulas

Probability Density Function

f(x)=Γ ⁣(ν+12)νπ Γ ⁣(ν2)(1+x2ν) ⁣−ν+12f(x) = \frac{\Gamma\!\left(\frac{\nu+1}{2}\right)}{\sqrt{\nu\pi}\,\Gamma\!\left(\frac{\nu}{2}\right)} \left(1 + \frac{x^2}{\nu}\right)^{\!-\frac{\nu+1}{2}}

Cumulative Distribution Function

F(x)=12+x Γ ⁣(ν+12)2F1 ⁣(12,ν+12;32;−x2ν)νπ Γ ⁣(ν2)F(x) = \frac{1}{2} + x\,\Gamma\!\left(\frac{\nu+1}{2}\right) \frac{{}_2F_1\!\left(\frac{1}{2},\frac{\nu+1}{2};\frac{3}{2};-\frac{x^2}{\nu}\right)}{\sqrt{\nu\pi}\,\Gamma\!\left(\frac{\nu}{2}\right)}

Properties

Mean

0  (ν>1)0 \;(\nu > 1)

Variance

νν−2  (ν>2)\frac{\nu}{\nu - 2} \;(\nu > 2)

Overview

Student's tt-distribution arises when estimating the mean of a normally distributed population with small sample sizes. It has ν\nu degrees of freedom and approaches the normal distribution as ν→∞\nu \to \infty.

Related Distributions

Normal Distribution Cauchy Distribution F-Distribution