The Student’s t Distribution

Let X ~ N(0, 1) and Yn ~ x2, where X and Yn are independent. Then the distribution of the random variable

VYn/n

is called the (Student’s2) t distribution with n degrees of freedom and is denoted

by tn.

exp(-(x[13] /n)y/2)
Vn/ yV2n

yn/2 1 exp(-y /2) _ Г(п/2)2”/2 ^

Подпись: TO hn (x) = j 0 Подпись: X

The conditional density hn (x |y) of Tn given Yn = y is the density of the N(1, n/y) distribution; hence, the unconditional density of Tn is

Г((п + 1)/2)

^ИЛГ(и/2)(1 + x 2/n)(n+1)/2

Подпись: var(Tn) = E[T2] Подпись: n n — 2 Подпись: (4.38)

The expectation of Tn does not exist if n = 1, as we will see in the next subsec­tion, and is zero for n > 2 by symmetry. Moreover, the variance of Tn is infinite for n = 2, whereas for n > 3,

See Appendix 4.A.

The moment-generating function of the tn distribution does not exist, but its characteristic function does, of course:

Подпись: Vtn(t)Г((п + 1)/2) f exp(it ■ x) ,

л/нжГ(п/2) j (1 + x2/n)(n+1)/2

— TO

TO

2 ■ Г((и + 1)/2W cos(t ■ x) ,

л/нжГ(п/2) j (1 + x2/n)(n+1)/2

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