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STAT 333 Assignment 2
Due: Friday, March 5 at the beginning of class
1.
Suppose we toss a fair coin repeatedly. Let
λ
be the event “H H T T ”.
a. Why is
λ
a renewal event?
b.
Use the renewal sequence {
r
n
} to show that
λ
is recurrent.
c. Determine
( )
R
s
λ
(the generating function of {
r
n
}), use it to obtain
and prove recurrence.
( )
Fs
d. Use
to calculate E[
T
( )
Fs
λ
]. Is
λ
positive recurrent or null recurrent?
e. Expand
in a power series and find
f
( )
Fs
8
, the probability that “H H T T”
first
occurs on
trial 8. Give a logical explanation for this probability.
f.
Use the Renewal Theorem to find E[
T
λ
]. Does this agree with the result in d)?
2. Suppose
λ
is a delayed renewal event.
a.
Prove the Delayed Renewal Relation:
()
()
()
Ds
Fs
R
s
=
. Hint: the proof is very similar to the
proof of the Renewal Relation in class. Justify your steps carefully.
b.
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This note was uploaded on 10/30/2011 for the course STAT 333 taught by Professor Chisholm during the Winter '08 term at Waterloo.
 Winter '08
 Chisholm
 Probability

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