Student's t-distribution STAT 587 (Engineering) Iowa State University March 30, 2021 (STAT587@ISU) Student's t-distribution March 30, 2021 1 / 13 Student's t distribution Probability density function Student's t distribution The random variable X has a Student's t distribution with degrees of freedom ν > 0 if its probability density function is ν+1 ν+1 2 − 2 Γ 2 x p(xjν) = p ν 1 + νπΓ( 2 ) ν where Γ(α) is the gamma function, Z 1 Γ(α) = xα−1e−xdx: 0 We write X ∼ tν. (STAT587@ISU) Student's t-distribution March 30, 2021 2 / 13 Student's t distribution Probability density function - graphically Student's t probability density function Student's t random variables 0.4 0.3 Degrees of freedom 1 0.2 10 100 0.1 Probablity density function, p(x) Probablity 0.0 −4 −2 0 2 4 x (STAT587@ISU) Student's t-distribution March 30, 2021 3 / 13 Student's t distribution Mean and variance Student's t mean and variance If T ∼ tv, then − ν+1 Z 1 ν+1 2 2 Γ 2 x E[X] = x p ν 1 + dx = ··· = 0; ν > 1 −∞ νπΓ( 2 ) ν and − ν+1 Z 1 ν+1 2 2 2 Γ 2 x ν V ar[X] = (x − 0) p ν 1 + dx = ··· = ; ν > 2: 0 νπΓ( 2 ) ν ν − 2 (STAT587@ISU) Student's t-distribution March 30, 2021 4 / 13 Student's t distribution Cumulative distribution function - graphically Gamma cumulative distribution function - graphically Student's t random variables 1.00 0.75 Degrees of freedom 1 0.50 10 100 0.25 Cumulative distribution function, F(x) distribution Cumulative 0.00 −4 −2 0 2 4 x (STAT587@ISU) Student's t-distribution March 30, 2021 5 / 13 Generalized Student's t distribution Location-scale t distribution Location-scale t distribution If X ∼ tν, then 2 Y = µ + σX ∼ tν(µ, σ ) for parameters: degrees of freedom ν > 0, location µ and scale σ > 0. By properties of expectations and variances, we can find that E[Y ] = µ, ν > 1 and ν V ar[Y ] = σ2; ν > 2: ν − 2 (STAT587@ISU) Student's t-distribution March 30, 2021 6 / 13 Generalized Student's t distribution Probability density function Generalized Student's t probability density function The random variable Y has a generalized Student's t distribution with degrees of freedom ν > 0, location µ, and scale σ > 0 if its probability density function is − ν+1 ν+1 2! 2 Γ 2 1 y − µ p(y) = ν p 1 + Γ( 2 ) νπσ ν σ 2 We write Y ∼ tν(µ, σ ). (STAT587@ISU) Student's t-distribution March 30, 2021 7 / 13 Generalized Student's t distribution Probability density function - graphically Generalized Student's t probability density function Student's t10 random variables 0.4 0.3 Scale, σ 1 2 0.2 Location, µ 0 0.1 2 Probablity density function, p(x) Probablity 0.0 −4 −2 0 2 4 x (STAT587@ISU) Student's t-distribution March 30, 2021 8 / 13 Generalized Student's t distribution t with 1 degree of freedom t with 1 degree of freedom 2 If T ∼ t1(µ, σ ), then T has a Cauchy distribution and we write T ∼ Ca(µ, σ2): If T ∼ t1(0; 1), then T has a standard Cauchy distribution. A Cauchy random variable has no mean or variance. (STAT587@ISU) Student's t-distribution March 30, 2021 9 / 13 Generalized Student's t distribution As degrees of freedom increases As degrees of freedom increases 2 If Tν ∼ tν(µ, σ ), then d 2 lim Tν = X ∼ N(µ, σ ) ν!1 (STAT587@ISU) Student's t-distribution March 30, 2021 10 / 13 Generalized Student's t distribution t distribution arise from a normal sample t distribution arising from a normal sample iid 2 Let Xi ∼ N(µ, σ ). We calculate the sample mean n 1 X X = X n i i=1 and the sample variance n 1 X S2 = (X − X)2: n − 1 i i=1 Then X − µ T = p ∼ t : S= n n−1 (STAT587@ISU) Student's t-distribution March 30, 2021 11 / 13 Generalized Student's t distribution Inverse-gamma scale mixture of a normal Inverse-gamma scale mixture of a normal If ν ν Xjσ2 ∼ N(µ, σ2=n) and σ2 ∼ IG ; s2 2 2 then 2 X ∼ tν(µ, s =n) which is obtained by Z 2 2 2 px(x) = pxjσ2 (xjσ )pσ2 (σ )dσ where px is the marginal density for x 2 pxjσ2 is the conditional density for x given σ , and 2 pσ2 is the marginal density for σ . (STAT587@ISU) Student's t-distribution March 30, 2021 12 / 13 Generalized Student's t distribution Summary Summary Student's t random variable: 2 T ∼ tν(µ, σ ); ν; σ > 0 E[X] = µ, ν > 1 ν 2 V ar[X] = ν−2 σ ; ν > 2 Relationships to other distributions (STAT587@ISU) Student's t-distribution March 30, 2021 13 / 13.