Page 242 - Elements of Distribution Theory

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228 Distribution Theory for Functions of Random Variables

It follows that

n n
r j

E a j X j |R 1 = r 1 ,..., R n = r n = a j
n + 1
j=1 j=1
so that

n n
1
E a j X j |R 1 ,..., R n = a j R j ;
n + 1
j=1 j=1
n
that is, E{ a j X j |R 1 ,..., R n } is a linear rank statistic.
j=1
7.7 Monte Carlo Methods
Let X denote a random variable, possibly vector-valued, with distribution function F X .
Suppose we are interested in the probability Pr(g(X) ≤ y) where g is a real-valued function
on the range of X and y is some specified value. For instance, this probability could represent
a p-value or a coverage probability. In this chapter, we have discussed several methods of
determining the distribution of Y = g(X). However, these methods often require substantial
mathematical analysis that, in some cases, is very difficult or nearly impossible.
Consider the following alternative approach. Suppose that we may construct a process
that generates data with the same distribution as X; let X 1 ,..., X N denote independent,
identically distributed random variables, each with the same distribution as X and let
Y j = g(X j ), j = 1,..., N.
Let
N
1
ˆ
P N = I {Y j ≤y}
N
j=1
denote the proportion of Y 1 ,..., Y N that are less than or equal to y. Thus, if N is large
enough, we expect that
ˆ
P N ≈ Pr(Y ≤ y).
ˆ
Hence, we use P N as an estimate of Pr(Y ≤ y). In fact, any type of statistical method, such
as a confidence interval, may be used to analyze the data generated in this manner.
This approach is known as the Monte Carlo method. The Monte Carlo method is a vast
topic. In this section, we give only a brief overview of the method; for further details, see
Section 7.9.

Example 7.29 (Ratio of exponential random variables to their sum). Let X = (X 1 , X 2 )
denote a random vector such that X 1 , X 2 are independent, identically distributed exponential
random variables with mean λ and let Y = X 1 /(X 1 + X 2 ); see Example 7.7. Consider the
probability Pr(Y ≤ 1/4) for λ = 1.
To estimate this probability, we can generate N pairs of independent standard exponential
random variables, (X 11 , X 21 ),..., (X 1N , X 2N ), and define

X 1 j
Y j = , j = 1,..., N.
X 1 j + X 2 j