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4. Functions of Random Variables and Sampling Distribution 207

particularly the normal, gamma, Chi-square, Student’s t, the F and beta distri-
butions. At this point, it will help to remind ourselves some of the techniques
used earlier to find the moments of normal, Chi-square or gamma variates.
In this section, we explain how one constructs the Student’s t and F vari-
ables. Through examples, we provide some justifications for the importance
of both the Student’s t and F variates. When we move to the topics of statis-
tical inference, their importance will become even more convincing.

4.5.1 The Student’s t Distribution
Definition 4.5.1 Suppose that X is a standard normal variable, Y is a Chi-
square variable with í degrees of freedom, and that X and Y are independently
distributed. Then, the random variable is said to have the
Student’s t distribution, or simply the t distribution, with í degrees of freedom.
Theorem 4.5.1 The pdf of the random variable W mentioned in the Defi-
nition 4.5.1, and distributed as t , is given by
í

with
Proof The joint pdf of X and Y is given by

for –∞ < x < ∞, 0 < y < ∞ where Let us denote u
= y and so that the inverse transformation is given by
and y = u. Note that From (4.5.1), the joint pdf
of U and W can be written as

for 0 < u < ∞, –∞ < w < ∞. Thus, for –∞ < w < ∞ and with the substitution s
= u(1 + w ν )/2, the pdf h(w) of W is given by
2 –1

which matches with the intended result. Some of the details are left out as the
Exercise 4.5.1.¢