Applied Statistics and Probability for Engineers

8-3

Now the numerator of Equation S8-1 is a standard normal random variable. The ratio in

the denominator is a chi-square random variable with n1 degrees of freedom, divided by

the number of degrees of freedom*. Furthermore, the two random variables in Equation

S8-1, Xand , are independent. We are now ready to state and prove the main result.S

S^2 ^2

Let Zbe a standard normal random variable and Vbe a chi-square random variable with

kdegrees of freedom. If Zand Vare independent, the distribution of the random variable

is the t-distribution with kdegrees of freedom. The probability density function is

f 1 t 2 

31 k (^12)  24
2 k 1 k 22
1
31 t^2 k 2 141 k^12 ^2
, t
T
Z
2 Vk
Theorem S8-1:
The
t-Distribution
Proof Since Zand Vare independent, their probability distribution is
Define a new random variable UV. Thus, the inverse solutions of
and
are
and
The Jacobian is
J †A
u
k
t
2 uk
01
†A
u
k
vu
zt
A
u
k
uv
t
z
1 vk
fZV 1 z, v 2 
v^1 k^22 ^1
22  2 k^2 a
k
2
b
e 1 z
(^2)  (^2)  2

, z, 0 

*We use the fact that follows a chi-square distribution with n1 degrees of freedom in Section 8-4

to find a confidence interval on the variance and standard deviation of a normal distribution.

1 n 12 S^2 ^2

PQ220 6234F.CD(08) 5/15/02 6:38 PM Page 3 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH114 FIN L:Quark F