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(x|ν) = √ ν 1 + νπΓ( 2 ) ν

where Γ(α) is the gamma function, Z ∞ Γ(α) = xα−1e−xdx. 0

We write X ∼ tν.

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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

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If T ∼ tv, then

− ν+1 Z ∞ ν+1
 2  2 Γ 2 x E[X] = x √ ν 1 + dx = ··· = 0, ν > 1 −∞ νπΓ( 2 ) ν and

− ν+1 Z ∞ ν+1
 2  2 2 Γ 2 x ν V ar[X] = (x − 0) √ 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

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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

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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 √ 1 + Γ( 2 ) νπσ ν σ

2 We write Y ∼ tν(µ, σ ).

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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

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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→∞

(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 = √ ∼ 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  X|σ2 ∼ 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) = px|σ2 (x|σ )pσ2 (σ )dσ

where

px is the marginal density for x 2 px|σ2 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

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