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4. Functions of Random Variables and Sampling Distribution 223

4.7.3 The Student’s t Distribution
In the Definition 4.5.1, we introduced the Student’s t distribution. In defining
W, the Student’s t variable, how crucial was it for the standard normal vari-
able X and the Chi-square variable Y to be independent? The answer is: inde-
pendence between the numerator and denominator was indeed very crucial.
We elaborate this with the help of two examples.
Example 4.7.4 Suppose that Z stands for a standard normal variable and
we let X = Z, Y = Z . Clearly, X is N(0, 1) and Y is but they are dependent
2
random variables. Along the line of construction of the random variable W, let
us write and it is clear that W can take only two
values, namely –1 and 1, each with probability 1/2 since P(X > 0) = P(X ≤ 0)
= 1/2. That is, W is a Bernoulli variable instead of being a t variable. Here, W
has a discrete distribution! !
Example 4.7.5 Let X , X be independent, X being N(0, 1) and X being
1
1
2
2
. Define X = X , , and obviously we have X < Y w.p. 1, so
2
1
that X and Y are indeed dependent random variables. Also by choice, X is N(0,
1) and Y is . Now, consider the Definition 4.5.1 again and write
1/2
so that we have | W |≤ (n + 1) . A random variable W with such a restricted
domain space can not be distributed as the Student’s t variable. The domain
space of the t variable must be the real line, ℜ. Note, however, that the ran-
dom variable W in (4.7.4) has a continuous distribution unlike the scenario in
the previous Example 4.7.4. !

4.7.4 The F Distribution

In the Definition 4.5.2, the construction of the F random variable was ex-
plained. The question is whether the independence of the two Chi-squares,
namely X and Y, is essential in that definition. The answer is “yes” as the
following example shows.
Example 4.7.6 Let U , U be independent, U be and U be . Define
2
1
2
1
X = U , Y = U + U , and we note that Y > X w.p.1. Hence, obviously X,Y are
1 1 2
dependent random variables. Also, X is and Y is . Now, we look at
the Definition 4.5.2 and express U as
so that we have 0 < U < (m + n)/m. A random variable U with such a
restricted domain space cannot have F distribution. The domain space