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Midterm 2 Review
1. (Stat 333 Winter 2010) Suppose a monkey randomly and independently hits the white
notes on a piano repeatedly, one at a time. The notes are
{
A,B,C,D,E,F,G
}
.
(a) Find the expected number of trials until the monkey ﬁrst plays the melody ”C D C
D E D C D C”.
(b) (Additional Part) Using the delayed renewal theorem, ﬁnd the expected number
of trials until the monkey ﬁrst plays the melody ”C C D C C A C C D C C”. (For
practice, ﬁnd the p.g.f.
F
λ
(
z
)
of the ﬁrst occurrence of this event and show that
F
0
λ
(0)
gives the same result.)
(c) Derive the p.g.f,
F
CDC
(
z
)
of the waiting time until the monkey ﬁrst plays ”C D C”.
(d) Find the probability that ”C D C” occurs for the ﬁrst time on trial 6.
(e) What two things are required for a delayed renewal event to occur inﬁnitely often?
(f) (Additional Part) Derive the p.m.f. of
T
CC
, the waiting time until ”CC” ﬁrst occurs.
2. (Stat 333 Winter 2010) Consider a Markov Chain with state space
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 Fall '08
 Chisholm
 Probability

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