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4. Functions of Random Variables and Sampling Distribution 233

transformation from Z to Y is not one-to-one. Follow along the Example 4.4.3.}
4.4.4 (Example 4.4.3 Continued) Suppose that Z has the standard normal
distribution. Let us denote Y = |Z| . Find the pdf of Y. {Caution: The transfor-
3
mation from Z to Y is not one-to-one. Follow along the Example 4.4.3.}
4.4.5 (Example 4.4.4 Continued) Suppose that X has the pdf f(x) = 1/
2exp{– |x|}I(x ∈ ℜ). Obtain the pdf of Y = |X| . {Caution: The transformation
3
from X to Y is not one-to-one. Follow along the Example 4.4.4.}
4.4.6 (Example 4.4.7 Continued) Verify all the details.
4.4.7 (Example 4.4.8 Continued) Verify all the details.
4.4.8 Suppose that X , ..., X are iid Uniform(0, 1). Let us define Y = X /
n
n:i
i
1
X n:i–1 , i = 1, ..., n with X = 1. Find the joint distribution of Y , ..., Y . {Hint:
2
n
n:0
First, consider taking log and then use the results from Examples 4.4.2 and
4.4.10.}
2
4.4.9 (Example 4.4.9 Continued) Suppose that X , ..., X are iid N(µ, σ )
1
n
with n ≥ 2. Use the Helmert variables Y , ..., Y from the Example 4.4.9 to
n
1
show that ( – 1) and S + S are independently distributed.
2
2/3
4.4.10 Suppose that (U, V) has the following joint pdf:
where θ > 0. Define X = UV and Y = U/V.
(i) Find the joint pdf of (X, Y);
(ii) Find the marginal pdf’s of X, Y;
(iii) Find the conditional pdf of Y given X = x.
4.4.11 Let X , X be iid having the Gamma(α, β) distribution with α > 0, β
2
1
> 0. Let us denote U = X + X , V = X /X . Find the marginal distributions of
2
2
1
1
U, V.
2
4.4.12 Let X , X be iid N(µ, σ ). Define U = X + X , V = X – X .
1 2 1 2 1 2
(i) Find the joint pdf of (U, V);
(ii) Evaluate the correlation coefficient between U and V;
2
(iii) Evaluate the following: E{(X – X ) &pipe; X + X = x}; E{(X 1
2
1
1
2
2
2
+ X ) | X = X }; V{(X – X ) | X + X = x}.
1
2
2
2
1
1
2
4.4.13 Suppose that X has the following pdf with –∞ < µ < ∞, 0 < µ < ∞: