Page 248 - Elements of Distribution Theory

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234 Distribution Theory for Functions of Random Variables

Define a random variable Z as follows. If Y 1 ≤ p(X 1 ), then Z = X 1 . Otherwise, if Y 2 ≤ p(X 2 ),
then Z = X 2 . Otherwise, if Y 3 ≤ p(X 3 ), then Z = X 3 , and so on. That is, Z = X j where
j = min{i: Y i ≤ p(X i )}.

(a) Show c > 1.
(b) Find the probability that the procedure has not terminated after n steps. That is, find the
probability that Z = X j for some j > n. Based on this result, show that the procedure will
eventually terminate.
(c) Find the distribution function of Z.
7.31 Let X denote a random variable with an absolutely continuous distribution with density func-
tion p and suppose that we want to estimate E[h(X)] using Monte Carlo simulation, where
E[|h(X)|] < ∞. Let Y 1 , Y 2 ,..., Y n denote independent, identically distributed random vari-
ables, each with an absolutely continuous distribution with density g. Assume that the distribu-
tions of X and Y 1 have the same support. Show that

1 n p(Y j )
E h(Y j ) = E[h(X)].
n g(Y j )
j=1
This approach to estimating E[h(X)] is known as importance sampling;a well-chosen
density g can lead to greatly improved estimates of E[h(X)].

7.9 Suggestions for Further Reading
The problem of determining the distribution of a function of a random variable is discussed in many
books on probability and statistics. See Casella and Berger (2002, Chapter 2) and Woodroofe (1975,
Chapter 7) for elementary treatments and Hoffmann-Jorgenson (1994, Chapter 8) for a mathematically
rigorous, comprehensive treatment of this problem.
Order statistics are discussed in Stuart and Ord (1994, Chapter 14) and Port (1994, Chapter 39).
There are several books devoted to the distribution theory associated with order statistics and ranks;
see, for example, Arnold, Balakrishnan, and Nagaraja (1992) and David (1981).
Monte Carlo methods are becoming increasingly important in statistical theory and methods.
Robert and Casella (1999) gives a detailed account of the use of Monte Carlo methods in statistics;
see also Hammersley and Handscomb (1964), Ripley (1987), and Rubinstein (1981).