Page 327 - Applied Statistics And Probability For Engineers

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8-3

2
Now the numerator of Equation S8-1 is a standard normal random variable. The ratio S 2 in
the denominator is a chi-square random variable with n 1 degrees of freedom, divided by
the number of degrees of freedom*. Furthermore, the two random variables in Equation
S8-1, X and , are independent. We are now ready to state and prove the main result.
S

Theorem S8-1:
The Let Z be a standard normal random variable and V be a chi-square random variable with
t-Distribution
k degrees of freedom. If Z and V are independent, the distribution of the random variable

Z
T
2V k
is the t-distribution with k degrees of freedom. The probability density function is

31k 12 24 1
f 1t2 2 1k 12 2 , t
2 k
1k 22 31t k2 14

Proof Since Z and V are independent, their probability distribution is

v 1k 22 1 2
f 1z, v2 e 1z 2 2 , z , 0

k
ZV
k 2

22 2
a b
2
Define a new random variable U V. Thus, the inverse solutions of
z
t
1v k
and

u v
are
u
z t
A k
and
v u

The Jacobian is
u t u
J † A k 2uk † A k
0 1

* We use the fact that 1n 12S 2 follows a chi-square distribution with n 1 degrees of freedom in Section 8-4
2
to find a confidence interval on the variance and standard deviation of a normal distribution.