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Unformatted text preview: IE 336
Oct. 7, 2009
Test #1
1. Let
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(b) Are the random variables X and Y independent? Explain.
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T“— ;ete'w i w—e' ~t // Name: ______ __ IE 336 Oct. 7, 2009 Name; _Som look Lu/ 2. Let X be a stationary Markov chain with states {2, 5, 9}. Let 2 dr pg
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P—Tgit‘f 2P3 5 be the onestep transition matrix of the Markov chain X (rows and columns of matrix P
are ordered in the same way the states are, i.e. the ﬁrst row/ column corresponds to state
2, the second row/ column corresponds to state 5, and the third row/ column corresponds to
state 9). Clearly p52 = P(X1 = 2Xo = 5) = 7", p29 = P(X1 = 9Xo = 2) = p, etc. a If q + ’I‘ = E compute p, q, and r and determine the transition diagram of this Markov
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0 am. (b) Find P(X5 = 9X4 = 5), P(Xg = 2X7 = 2,X2 = 5), and P(X3 = 5X4 = 5,955 = 9).
(C) DGteTmine the walk PIObabilitieS P2259225952, 135922595225, p2259522592, and 109522592259 (d) Assume that at some point the process is in state 5. Compute the expected number of
visits to state 2 in the following three steps. (e) Assume that at time step k + 2 the process is three times more likely to be in state 5 than in state 9. Also, assume that at time step k + 5 the process is in state 9 with L_
probability 0.5. Find the probability that at time step k + 4 the process is in state 2. 0 3
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9
Oct. 7, 2009 Name: 3. In this problem we observe how the weather changes on a day—today basis in Indiana in
late fall. We assume that observing the weather outside each day at 3pm is a good indication
how the weather was during the entire day. Further, for simplicity we will assume that our
weather observations can be: 1) sunny, 2) cloudy, and 3) raining. After many years of
gathering statistical data we ﬁnally got to know that the weather is never the same two
days in a row (what a surprise!) Also, we know that if one day the weather is sunny then
with probability % the following day will be cloudy. On the other hand, if one day is cloudy
the following day is two times more likely to be raining than sunny and if one day is raining
the following day is three times less likely to be sunny than cloudy. Assume that these
weather changes can be modeled as a stationary Markov chain. (a) Determine the transition diagram and the transition matrix of this Markov chain. (1)) If the weather is sunny on Sunday, determine the probability that it will be sunny on
the ﬁrst coming Wednesday and Saturday. (0) If it is raining on Tuesday afternoon, ﬁnd the average number of sunny days that we
will have before Friday 10pm. ((1) If on a randomly chosen day it is three times less likely to be sunny than cloudy and
two times more likely to be cloudy than raining ﬁnd the probability that three days
later the weather is sunny. 3 c R
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