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202 4. Functions of Random Variables and Sampling Distribution
Example 4.4.11 (Example 4.4.10 Continued) Suppose that the Xs are iid
and the common pdf is the same as the one in (4.4.11). For all k = 1, ..., n
1, observe that
z
and hence
Since we have
It is worthwhile to note that we have succeeded in deriving an expression for
the expected value of the k order statistic X without finding the marginal
th
n:k
distribution of X . These techniques are particularly useful in the areas of
n:k
reliability and survival analyses. !
If the Xs are iid normal, the Helmert transformation provides
a natural way to consider intricate properties of and S . If the
2
Xs are iid exponential or negative exponential, the transformation
involving the spacings between the successive order statistics
is a natural one to consider instead.
Example 4.4.12 The Negative Exponential Distribution: Suppose that
X , ..., X are iid random variables having a common pdf given by
1 n
where µ is the location parameter and σ is the scale parameter. In reliability
applications, µ is often referred to as the minimum guarantee time or the
minimum threshold, and hence µ is assumed positive in such applications.
Refer back to (1.7.36) as needed. We will, however, continue to assume that
µ is an arbitrary real number. Let us consider the two random variables
and look at their distributions. Denote W = X µ and then it is clear that
i
i
W , ..., W are iid having the common pdf given by in (4.4.11) with β
1
n
replaced by σ. Following the Example 4.4.10, we write U = W , U =
2
n:1
1
