Page 316 - Fundamentals of Probability and Statistics for Engineers

Parameter Estimation 299

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

n‡1†=2

‰ n ‡ 1†=2Š t 2

f T t†ˆ 1 ‡ ; 1 < t < 1: 9:134†

n=2† n † 1=2 n

This distribution is known as Student’s t-distribution with n degrees of freedom;
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

!
8
1 1
2
> u =2 n=2† 1 v=2
< e v e ; for 1 < u < 1; and v > 0;
f UV u; v†ˆ 2 † 1=2 2 n=2
n=2†
>
:
0; elsewhere:
9:135†
Consider the transformation from U and V to T and V. The method discussed
in Section 5.3 leads to
1
1
f TV t; v†ˆ f UV ‰g t; v†; g t; v†ŠjJj; 9:136†
1 2
where
v 1=2 1

1
g t; v†ˆ t ; g t; v†ˆ v; 9:137†
1 2
n
and the Jacobian is
1 1
qg qg
1

1
v 1=2 v 1=2 1
qt qv t 2n v 1=2
J ˆ ˆ n n ˆ : 9:138†
qg qg
1 1 n
2 2 0 1

qt qv
The substitution of Equations (9.135), (9.137), and (9.138) into Equation
(9.136) gives the jpdf f TV (t, v) of T and V. The pdf of T as given by Equation
(9.134) is obtained by integrating f TV (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.

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