hw8-sol - AMS 311(Fall 2009 Joe Mitchell PROBABILITY THEORY...

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AMS 311 (Fall, 2009) Joe Mitchell PROBABILITY THEORY Homework Set # 8 – Solution Notes (1). (16 points) Consider the maze shown below. There are three cells (Cell 1, Cell 2, and Cell 3) and two deadly (quite permanent) outcomes (Death By Poison, and the dreaded Death By Torture). A rat is initially placed in cell 3. When the rat enters Cell i , he wanders around within the cell for X i minutes, where X i has density f ( x ) = braceleftbigg 2 x if 0 x 1 0 otherwise, and then he exits the cell by picking one of the doors at random (if there are 3 doors, he picks each with probability 1/3). 3 1 2 Poison Torture (a). (9 points) Find the probability that the rat dies by poison. (Recall that he starts in Cell 3.) Let p i = P ( dies by poison | starts in cell i ). Condition on the first move: p 1 = (1 / 4)1 + (2 / 4) p 2 + (1 / 4) p 3 p 2 = (1 / 5)0 + (2 / 5) p 1 + (2 / 5) p 3 p 3 = (1 / 5)1 + (1 / 5) p 1 + (2 / 5) p 2 + (1 / 5)0 Solve to get p 1 , p 2 , and p 3 . The final answer is p 3 (since we know the rat starts in Cell 3). I calculated that p 3 = 25 / 51 (and p 1 = 10 / 17, p 2 = 22 / 51). (b). (9 points) What is the expected number of minutes that the rat lives? The expected time (in minutes) the rat spends in Cell i before leaving it is τ = integraldisplay 1 0 x · 2 xdx = 2 3 Let t i = E ( number of minutes he lives | start in cell i ). Condition on the first move: t 1 = (1 / 4)( τ + 0) + (2 / 4)( τ + t 2 ) + (1 / 4)( τ + t 3 ) t 2 = (1 / 5)( τ + 0) + (2 / 5)( τ + t 1 ) + (2 / 5)( τ + t 3 ) t 3 = (1 / 5)( τ + 0) + (1 / 5)( τ + t 1 ) + (2 / 5)( τ + t 2 ) + (1 / 5)( τ + 0) Solve for the t i ’s; the final answer is t 3 . (In each of the above problems, you should DEFINE a set of unknowns precisely, set up a system of equations involving the unknowns, and then solve for the final answer (time permitting). You need not solve the actual system, if you give a complete description of how you would finish solving the problem.) (2). (20 points) Let X have probability density function f ( x ) = braceleftBigg x/ 2 if 0 < x < 1 3 / 4 if 2 < x < 3 ; 0 otherwise.
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