Page 328 - Applied Statistics And Probability For Engineers

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8-4

Thus,

u
0J 0
A k

and the joint probability distribution of T and U is

1u 1k 22 1 31u k2 t u4 2
2
1t, u2 e
k
f TU u
k 2

12 k 2
a b
2
Now, since V 0, we must require that U 0, and since Z , T . On
rearranging this last equation, we have
1 1k 12 2 1u 22 31t k2 14
2
f 1t, u2 u e , 0 u , t
TU
k
22 k 2
a b
k 2

2
The probability distribution of T is found by

f 1t2 f 1t, u2 du
T TU
0

1 2
u 1k 12 2 1u 22 31t k2 14 du
e
k
k 2
22 k 2
a b 0
2

31k 12 24 1
2 1k 12 2 , t
k 31t k2 14
2 k
a b
2
This is the distribution given in Theorem S8-1.
The probability distribution of the random variable T was first published by W. S. Gosset
in a famous 1908 paper. Gosset was employed by the Guiness Brewers in Ireland. Since his
employer discouraged publication of employee research, Gosset published these results under
the pseudonym “Student.” As a result, this probability distribution is sometimes called the
Student t-distribution.