Fundamentals of Probability and Statistics for Engineers

If U is N(0, 1), V is^2 -distributed with n degrees of fr eedom, and U and V are
independent, then the pdf of T has the form

This distribution is known as Student’s t-distribution with n degrees of fr eedom;
it is named after W.S. Gosset, who used the pseudonym ‘Student’ in his
research publications.

Proof of Theorem 9.7:the proof is straightforward following methods given
in Chapter 5. Sine U and V are independent, their jpdf is

Consider the transformation from U and V to T and V. The method discussed
in Section 5.3 leads to

where

and the Jacobian is

The substitution of Equations (9.135), (9.137), and (9.138) into Equation
(9.136) gives the jpdf fTV(t,v) of T and V. The pdf of T as given by Equation
(9.134) is obtained by integrating fTV(t, v) with respect to v.

It is seen from Equation (9.134) that the t-distribution is symmetrical about
the origin. As n increases, it approaches that of a standardized normal random
variable.

Parameter Estimation 299

fT…t†ˆ

‰…n‡ 1 †= 2 Š

…n= 2 †…n†^1 =^2

1 ‡

t^2

n

…n‡ 1 †= 2

; 1<t< 1 : … 9 : 134 †

fUV…u;v†ˆ

1

… 2 †^1 =^2

eu

(^2) = 2
!
1
2 n=^2 …n= 2 †
v…n=^2 †^1 ev=^2

; for1<u< 1 ;andv> 0 ;
0 ; elsewhere:
8

<
9 : 135 †
fTV t;v†ˆfUV‰g 11 t;v†;g 21 t;v†ŠjJj; 9 : 136 †
g 11 t;v†ˆt
v
n

 1 = 2

; g 21 …t;v†ˆv; … 9 : 137 †

Jˆ

qg 11

qt

qg 11

qv

qg 21

qt

qg 21

qv

(^)
(^)

ˆ

v

n

 1 = 2

t 2 n

v

n

 1 = 2  1

01

ˆ

v

n

 1 = 2

: … 9 : 138 †