Stat 333  Assignment 2  Fall 2011
Due Thursday November 10 at the beginning of class
1. Consider repeated Bernoulli trials, each with a probability of success
P
(
S
) =
p
.
Let
e
n
=
Pr(even number of
S
’s in
n
trials). Assume zero is even.
(a) Show that
e
n
=
q
·
e
n

1
+
p
·
(1

e
n

1
)
,
n
≥
1
.
(b) Hence, find the probability generating function of
{
e
n
}
n
≥
0
.
(c) Let
p
= 0
.
5
. Find an explicit expression for
e
n
,
n
≥
0
.
2. Prove the Delayed Renewal Relation. (Hint: the proof is very similar to that of the Re
newal Relation, and you have to be very careful with the starting values of sums.)
3. Javid’s worldclass cricketing career has recently begun. In a single game of cricket, he
is said to score a
century
if he scores at least 100 runs. The number of matches until he
scores a century has a
Geometric
(0
.
01)
distribution.
(a) Let
λ
be the event that he scores a century in a game.
i. Explain carefully why
λ
is a renewal event.
ii. Determine the renewal sequence
{
r
n
}
.
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 Fall '08
 Chisholm
 Bernoulli, Probability, DNA, Probability theory, renewal sequence, delayed renewal

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