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Unformatted text preview: 1. Let X and Y be continuous random variables with joint probability dens1ty Iunctlon _ 3(x2+y2)/2 if0<:1:<1,0<y<1;
f ($:y)—{ 0 otherwise. Find E(XY). c 2.
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p 2. Taxi driver Jimmy plies his trade in two cities: Houston and Sugarland, indexed by 0 and 1, respectively. The transition probability matrix governing his movement is given 0.6 0.4
P ‘ [0.5 0.5]' For example, if the last trip ended in Houston then with probability 0.6 and 0.4, his as follows: next trip will end in Houston and Sugarland. In a long run, what are the proportions of Jimmy’s trips ended in Houston and Sugarland, respectively. SE = 0.6 m + 0‘5“ $7 1T0=
'[T‘ = 04 Tlo +0351.“
Tile 1H] : \ 3. A player throws a fair die and simultaneously ﬂips a fair coin. If the coin lands heads,
then she Wins twice of the values that appears on the die. If the coin lands tails, then
she wins half of the values that appears on the die. Determine her expected winnings. E mmmﬁs) = Air: (Winnings Hm ) + J;— Emnmj \ mi 5
, J7: (J—(Jz.\+.. 45.63) +47: (1(tl+..+%.63>  7.
1 325.
L4.%+ 2.  g H 4. Let the Markov chain consisting of the states 0, 1, 2, 3 and 4 have the transition probability matrix
0.5 0 0 0.5 0
0 O 1 0 0
P: 0 0 0 1 0
0 0 0 0 1
0 0 0.3 0.3 0.4 Determine the classes of the Markov chain. Determine which states are recurrent and which states are transient. 108 'chm9 lent 0 ——> 7)
I in} transient“
l——72 +_——4 §Z,'5,Ari Mumat Two white and four black balls are distributed in two urns in such a way that each
contains three balls. We say that the system is in state i, i = 0,1,2, if the ﬁrst urn
contains i white balls. At each step, we draw one ball from each um and the two selected balls are interchanged. Let X" denote the state of the system after the nth
step. Calculate its transition probability matrix. 0';
2.
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 Spring '08
 Whitt

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