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4. Functions of Random Variables and Sampling Distribution 239

pdf of T , ..., T , distributed as Mt (ν, ΣΣ ΣΣ Σ), is given by
p
1
p

with t’ = (t , ..., t ). {Hint: Start with the joint pdf of X , ..., X and Y. From
p
1
p
1
this, find the joint pdf of T , ..., T and Y by transforming the variables (x , ...,
1
p
1
x , y) to (t , ..., t , y). Then, integrate this latter joint pdf with respect to y.}
p 1 p
2
4.6.12 (Exercise 4.6.5 Continued) Let X , ..., X be iid N(µ, σ ). Recall
1
n
that Y = X – X , i = 2, ..., n. Suppose that U is distributed as and also
i
1
i
U is independent of the X’s. Find the joint pdf of Y /U for i = 2, ..., n.
i
4.6.13 Consider the independent random variables X , X , ..., X where X i
p
0
1
is distributed as . We denote the new random variables U =
i
(X /ν ) ÷ (X /ν ), i = 1, ..., p. Jointly, (U , ..., U ) is said to have the p-
p
0
1
0
i
i
dimensional F distribution, denoted by MF (ν , ν , ..., ν ). Using transforma-
0
p
p
1
tion techniques, show that the joint pdf of U , ..., U , distributed as MF (ν ,
p
0
1
p
ν , ..., ν ), is given by
1 p
with u’ = (u , ..., u ) and . {Hint: Start with the joint pdf of X ,
1
0
p
X , ..., X . From this, find the joint pdf of U , ..., U and X by transforming
1
p
1
p
0
(x , ..., x , x ) to (u , ..., u , x ). Then, integrate this latter pdf with respect to
0
p
p
1
0
1
x .}
0
4.6.14 Suppose that (U , U ) is distributed as N (5, 15, 8, 8, ρ) for some
2
2
1
ρ ∈ (–1, 1). Let X = U + U and X = U – U . Show that X and X are
1
2
2
1
2
1
1
2
independently distributed. {Hint: Is (X , X ) jointly distributed as a bivariate
2
1
normal random vector?}
4.7.1 (Examples 3.6.3 and 4.7.1 Continued) Suppose that a pair of random
variables (X , X ) has the joint pdf given by
1
2
with 0 < α < 1, where stands for the bivariate nor-
mal pdf defined in (3.6.1) with means µ , µ , variances , and the
2
1
correlation coefficient ρ with 0 < ρ < 1. Show that X + X is not normally
1 2