STAT 333  Spring 2009  Assignment 2
Due: Thursday, June 18 at 2:30 pm (in class)
Type I Problems
1. Suppose
X
has probability density function
f
(
x
) = 2
x

3
for
x
≥
1. (This is called a
powerlaw or Pareto distribution.) Suppose
Y
is independent of
X
and has a gamma
distribution with parameters
α
= 4 and
λ
= 2. Obtain
P
(
X > Y
).
2. #37, page 171 of the text, 9th edition.
3. #44, page 173 of the text, 9th edition.
4. #62, page 176 of the text, 9th edition.
Type II Problems
1. For each of the following distributions, find the pgf
G
X
(
s
), find the radius of conver
gence, and use it to find
E
[
X
] and Var(
X
).
(a)
X
is a Poisson random variable with parameter
λ
.
(b)
X
is a discrete uniform random variable over the integers 1, 2, ...,
k
1,
k
. That
is,
P
(
X
=
i
) = 1
/k
for each of
i
= 1, ...,
k
.
2. Suppose we toss a fair coin repeatedly and observe a sequence of H or T. Let
λ
be
the event ”T T H H”.
(a) Why is
λ
a renewal event?
(b) Use the renewal sequence
r
n
to show that
λ
is recurrent.
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 Spring '08
 Chisholm
 Probability, Probability theory, fλ, renewal sequence, delayed renewal, Delayed Renewal Relation

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