20082ee131A_1_HW8

# 20082ee131A_1_HW8 - + 1) = n ! . 3. Toss a fair 3-sided die...

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EE 131A Homework #8 Spring 2008 Due May 28th K. Yao Read Leon-Garcia (3rd edition), pp.256-284; 67-70; 369-375 . 1. A miner is trapped in a mine containing 3 doors. The ﬁrst door leads to a tunnel that will take him to safety after 3 hours of travel. The second door leads to a tunnel that will return him to the mine after 5 hours of of travel. The third door leads to a tunnel that will return him to the mine after 7 hours of travel. If we assume the miner is all times equally likely to chose any one of the door, what is the expected length of time he reaches safety? Hint: Let X denote the amount of time (in hours) until the miner reaches safety and let Y denote the door he initially chooses. Find E { X | Y = y j } and then use E { X } = J j =1 E { X | Y = y j } P ( Y = y j ) . 2. The r.v. X and Y have a joint pdf given by f X,Y ( x,y ) = 2 e - 2 x /x, 0 < x < , 0 y x ; f X,Y ( x,y ) = 0 , otherwise . Compute cov ( X,Y ) . Hint: R 0 y n e - y dy = Γ( n
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Unformatted text preview: + 1) = n ! . 3. Toss a fair 3-sided die with face values of { 1 , 2 , 3 } . Let N be a rv denoting the face value of the die. If N = n, n = 1 , 2 , 3 , pick a continuous rv X with a uniform distribution between 0 and n. a. Find f X ( x ) , f N ( n ) , f N,X ( n,x ) , f N | X ( n | x ) , and f X | N ( x | n ) . b. Consider f N | X ( n | x ) . Decide what is supposed to add up or integrate to 1 for it to be a proper conditional pdf. c. Find E { X } . d. Find P (0 X 1) . e. Find E ( N | X = x ) . f. Find P ( N = 2 | X = x )) . 4. Do Problem 5.11 (p. 289, 3rd edition of the textbook). 5. Do Problem 5.12 (p. 289, 3rd edition of the textbook). 6. Do Problem 5.18 (p. 290-291, 3rd edition of the textbook). 7. Do Problem 5.77 (p. 296, 3rd edition of the textbook)....
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## This note was uploaded on 04/12/2010 for the course EE 131A taught by Professor Lorenzelli during the Spring '08 term at UCLA.

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