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4. Functions of Random Variables and Sampling Distribution 201

and other historical notes, one may refer to Zinger (1958), Lukacs (1960),
and Ramachandran (1967, Section 8.3), among a host of other sources. Look
at the Exercises 4.4.22-4.4.23 which can be indirectly solved using this char-
acterization.
Example 4.4.10 The Exponential Spacings: Let X , ..., X be iid with the
n
1
common pdf given by

Define the order statistics X ≤ X ≤ ... ≤ X and let us write Y = X , i =
n:2
n:n
i
n:i
n:1
1, ..., n. The X ’s and X ’s may be interpreted as the failure times and the
n:i
i
ordered failure times respectively. Let us denote
and these are referred to as the spacings between the successive order statis-
tics or failure times. In the context of a life-testing experiment, U corre-
1
sponds to the waiting time for the first failure, and U corresponds to the time
i
between the (i – 1) and the i failures, i = 2, ..., n. Here, we have a one-to-
th
th
one transformation on hand from the set of n variables (y , ..., y ) to another
1
n
set of n variables (u , ..., u ). Now, in view of (4.4.4), the joint pdf of the
1
n
order statistics Y , ..., Y is given by
1 n
Note that . One can
also verify that | det(J) |= 1, and thus using (4.4.4) and (4.4.13), the joint pdf
of U , ..., U can be written as
1 n

for 0 < u , ..., u < ∞. Next, (4.4.14) can be easily rewritten as
n
1

for 0 < u , ..., u < ∞. In (4.4.15), the terms involving u , ..., u factorize and
n
1
n
1
none of the variables’ domains involve any other variables. Thus it is clear
(Theorem 3.5.3) that the random variables U , ..., U are independently dis-
n
1
tributed. Using such a factorization, we also observe that U has an exponen-
i
–1
tial distribution with the mean (n – i + 1) β,i = 1, ..., n. !