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7.8 Exercises 231

Carlo analysis are very useful as a check on the theoretical calculations. When approxima-
tions to Pr(Y ≤ y) are used, as will be discussed in Chapters 11–14, the results of a Monte
Carlo study give us a method of assessing the accuracy of the approximations.
The importance of the drawbacks discussed above can be minimized by more sophis-
ticated Monte Carlo methods. For instance, many methods are available for reducing the
variation in the Monte Carlo results. Also, before carrying out the Monte Carlo study, it is
important to do a thorough theoretical analysis. For instance, in Example 7.34, even if the
distribution of X 1 /(X 1 + X 2 )is difficult to determine analytically, it is easy to show that
the distribution of this ratio does not depend on the value of λ; thus, the results for λ = 1
can be assumed to hold for all λ> 0.

7.8 Exercises
7.1 Let X denote a random variable with a uniform distribution on the interval (0, 1). Find the density
function of
X
Y = .
1 − X
7.2 Let X denote a random variable with a standard normal distribution. Find the density function of
Y = 1/X.
7.3 Let X denote a random variable with a Poisson distribution with mean 1. Find the frequency
function of Y = X/(1 + X).
7.4 Let X denote a random variable with an F-distribution with ν 1 and ν 2 degrees of freedom. Find
the density function of

ν 1 X
Y = .
ν 2 1 + (ν 1 /ν 2 )X
7.5 Let X 1 and X 2 denote independent, real-valued random variables with absolutely continuous
distributions with density functions p 1 and p 2 , respectively. Let Y = X 1 /X 2 . Show that Y has
density function
∞

p Y (y) = |z|p 1 (zy)p 2 (z) dz.
−∞
7.6 Let X 1 , X 2 denote independent random variables such that X j has an absolutely continuous
distribution with density function

λ j exp(−λ j x), x > 0,
j = 1, 2, where λ 1 > 0 and λ 2 > 0. Find the density of Y = X 1 /X 2 .
7.7 Let X 1 , X 2 , X 3 denote independent random variables, each with an absolutely continuous distri-
bution with density function
λ exp{−λx}, x > 0

where λ> 0. Find the density function of Y = X 1 + X 2 − X 3 .
7.8 Let X andY denoteindependentrandomvariables,eachwithanabsolutelycontinuousdistribution
with density function
1
p(x) = exp{−|x|}, −∞ < x < ∞.
2
Find the density function of Z = X + Y.