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HW7_solutions(1)

# HW7_solutions(1) - HW#7 Solutions AMS 311 Probability...

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HW#7, Solutions AMS 311 Probability Theory (Spring 2011) Part 1 (10 points): PRINT your name and StonyBrook ID ( Last, First, #ID ) on the top on your work sheet. Staple your HW if more than 1 pages. Part 2: Problems (90 points) Write down your work step by step to get full score. (1) (30 points) (a) Denote p i =P(the rat dies by poison | the rat starts in Cell i ). Then conditioning on the first door chosen, we get: p 1 =(1/5)*0+(2/5)*1+(2/5)p 3 p 2 =1*p 3 p 3 =(1/5)*1+(2/5)*p 1 +(2/5)*p 2 Solve, getting p 1 =8/11, p 2 =p 3 =9/11. (b) Let Y be the number of minutes the rat lives in total. Let t i = E(Y | the rat starts in Cell i). We want to compute t 1 . Note that the rat wanders in Cell i for an expected amount of time ( i+i 2 )/2 minutes, since the expected value of a Uniform( i, i 2 ) is ( i+i 2 )/2. Then, conditioning on the first door chosen, we get: t 1 =(1+1 2 )/2+(1/5)*0+(2/5)*0+(2/5)*t 3 t 2 =(2+2 2 )/2+1*t 3 t 3 =(3+3 2 )/2+(1/5)*0+(2/5)*t 1 +(2/5)*t 2 Solve, we get t 1 =87/11 mins (c) Let q i = P( the rat visits Cell 2 before he dies | the rat starts in Cell i). Then, conditioning on the first door chosen, we get: q 1 =(1/5)*0+(2/5)*0+(2/5)*q 3 q 2 =1 q 3 =(1/5)*0+(2/5)*q 1 +(2/5)*q 2 Solve, we get q 1 =4/21 (2). (30 points) Homework is due at the beginning of class on its due date. You must be in class and on time to submit your homework. No homework will be accepted via email. No late homework will be accepted. Your lowest homework grade will be dropped before computing your average.

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HW#7, Solutions
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HW7_solutions(1) - HW#7 Solutions AMS 311 Probability...

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