Stat 150 Midterm, J.P., Spring 2006. Student name in CAPITALS:
A. Warmup. Little explanation required. Just apply results from class notes or text.
1. A Poisson(
λ
) number of dice are rolled. Let
N
i
be the number of times face
i
appears among
these dice. Describe the joint distribution of
N
1
and
N
2
.
2. A random walk starting at 3 moves up 1 at each step with probability 2
/
3 and down 1 with
probability 1
/
3. What is the probability that the walk reaches 10 before 0?
3. For the same random walk, what is the expected number of steps until reaching either 0 or
10?
4. Let
T
r
be the number of tosses until the
r
th H in independent coin tosses with
p
the probability
of H on each toss. Write down the probability generating function of
T
r
.
5. Let
U
r
be the number of tosses until the pattern of ”HTHT
....
HT” of length 2
r
appears
(meaning the ”HT” is repeated
r
times in a row). Give a formula for
E
(
U
r
).
B. Consider a Markov chain (
X
n
) with transition matrix
P
. Let
f
be a function with numerical
values defined on the state space of the Markov chain.
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 Spring '09
 ODEAN
 Probability, Probability theory, Markov chain, independent coin tosses

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