hw4sol - UCSD ECE 153 Prof. Young-Han Kim Handout #16...

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UCSD ECE 153 Handout #16 Prof. Young-Han Kim Thursday, October 30, 2008 Solutions to Homework Set #4 (Prepared by TA Halyun Jeong) 1. Read Sections 4.3, 4.6, 5.7, 5.9, 6.5 in the text. Try to work on all examples. 2. Two envelopes. An amount A is placed in one envelope and the amount 2 A is placed in another envelope. The amount A is fixed but unknown to you. The envelopes are shuffled and you are given one of the envelopes at random. Let X denote the amount you observe in this envelope. Designate by Y the amount in the other envelope. Thus ( X, Y ) = ( ( A, 2 A ) , with probability 1 2 , (2 A, A ) , with probability 1 2 . You may keep the envelope you are given, or you can switch envelopes and receive the amount in the other envelope. (a) Find E ( X ) and E ( Y ). (b) Find E ± X Y ) . (c) Suppose you switch. What is the expected amount you receive? Solution: (a) The expected amount in the first envelope is E ( X ) = X x ∈X xp X ( x ) = 1 2 A + 1 2 (2 A ) = 3 2 A. Since you are given one of the envelopes at random, the expectation is the same for the envelope you are not given. Thus the expected amount in the second envelope is E ( Y ) = 3 2 A. (b) The expected factor by which the amount in the second envelope exceeds the amount in 1
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E ± Y X = X ( x,y ) ∈X×Y y x · p XY ( x, y ) = 1 2 A 2 A + 1 2 2 A A = 1 4 + 1 = 5 4 . (c) If you switch, the expected amount you will receive is E ( Y ) = 3 2 A . 3. Tall trees. Suppose that the average height of trees on campus is 20 feet. Argue that no more than half of the tree population is taller than 40 feet. Solution: The average height of the trees in the population is 20 feet. So 1 n n i =1 h i = 20, where n is the population size and h i is the height of the i -th tree. If more than half of the population is at least 40 feet tall, then the average will be greater than 1 2 · 40 = 20 feet since each person is at least 0 feet tall. Thus no more than half of the population is 40 feet tall. Alternatively, we can use the Markov inequality with respect to the fraction of population to obtain the same result. 4. Let Λ and X be two random variables with Λ f Λ ( λ ) = 5 3 λ 2 3 , 0 λ 1 0 , otherwise, and X |{ Λ = λ } ∼ Exp( λ ). Find E ( X ). Solution:
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This note was uploaded on 10/26/2011 for the course MATH 180C taught by Professor Eggers during the Winter '09 term at Aarhus Universitet.

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hw4sol - UCSD ECE 153 Prof. Young-Han Kim Handout #16...

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