STAT 510 HW #5 – Written
Due Friday, October 6, 2006
You can turn it in in class, or to my office (116B IH) or mailbox in 101 IH, by 4PM.
There are also questions on Mallard.
1. Suppose Z∼N(0,1) and U∼U niform(0,1), and Zand Uare independent. Let
Y= (Y1, Y2) = g(z, u) be given by
Y1=Z
Uand Y2=U.
Y1is said to have the “slash” distribution.
(a) What is the space of Y?
(b) Find g−1(y).
(c) Find Jg−1(y).
(d) Find the pdf of Y.
(e) Are Y1and Y2independent? Why or why not?
(f) What is the marginal space of Y1? Show that the marginal pdf of Y1is
f1(y1) = c. 1−e−y2
1/2
y2
1
.
What is the constant c?
2. Suppose (Y1, . . . , Y5)∼Dirichlet(α1, . . . , α6). Find the distribution of
(U1, U2) = (Y1+Y2, Y3+Y4+Y5).
(Do not use pdf’s, unless you really want to. Work directly with the definition of the Dirichlet
based on gamma’s.)
4. Suppose (Y1, Y2, Y3)∼Dirichlet(α1, α2, α3, α4). Let U=g(Y), where
U1=Y1, U2=Y1+Y2, U3=Y1+Y2+Y3.
(a) Find the matrix Afor which U=AY.
(b) Find Jg−1(u).
(c) Find the pdf of U.
1