HW3 - Mathematics Department, UCLA T. Richthammer winter...

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Unformatted text preview: Mathematics Department, UCLA T. Richthammer winter 09, sheet 3 Jan 16, 2009 Homework assignments: Math 170A Probability, Sec. 1 025. Suppose you perform the following experiment: First you roll a fair die. If the die shows the value n , then you choose a number from { 1 , . . . , n } completely at random. (a) Define a probabilistic model for this situation and define RVs describing the value of the die and of the number chosen. Sketch the corresponding tree diagram. (Hint: For describing the model assume that the number chosen can in principle take all values from { 1 , . . . , 6 } , but some of them only with probability 0.) (b) For m = 1 , . . . , 6 calculate the probability that the number chosen is m . (c) What is the probability that the number chosen is the value of the die? Answer: (a) S = { 1 , . . . , 6 } 2 , p i = 1 6 and p j | i = 1 i if j ≤ i and p j | i = 0 if j > i . This defines a sequential model. The projections X 1 , X 2 represent the value of the die and the number chosen. For the tree diagram: picture ... (b) P ( X 2 = m ) = 6 ∑ i =1 P (( i, m )) = 6 ∑ i =1 p i p m | i = 1 6 6 ∑ i =1 p m | i = 1 6 6 ∑ i = m 1 i , and calculating this for m = 1 , . . . , 6 we get m 1 2 3 4 5 6 P ( X 2 = m ) 147 360 87 360 57 360 37 360 22 360 10 360 . (c) P ( X 1 = X 2 ) = 6 ∑ i =1 P (( i, i )) = 6 ∑ i =1 p i p i | i = 1 6 6 ∑ i =1 1 i = 147 360 . 026. Suppose you have n coins. Every coin has small imperfections in the distribution of the metal, so probably none of them is fair. Let q ( i ) be the probability for Head for the i th coin. We assume that for large n the imperfections average out, i.e. q (1)+ ... + q ( n ) n = 1 2 . A “fair random generator” is supposed to be a device that gives you two possible outputs with probability 1/2 each. (a) Using the n coins, describe (in words) a sequential experiment that acts as a fair random generator. (Hint: See the corresponding example of the lecture.) (b) Define the corresponding sequential model, and define two events you want to use as outputs. Check by a calculation that these outputs have probability 1/2. (c) Draw a tree diagram describing your experiment. Answer: (a) Choose one of the n coins at random and toss it. Whatever the tossed coin shows is the output of a fair random generator. (b) S = { 1 , . . . , n }×{ , 1 } , where ( i, j ) represents the outcome that coin i was chosen, and that the chosen coin shows Head (for j = 1) or Tail (for j = 0) respectively. Let X 1 , X 2 be the projections. Let p i = 1 n , p 1 | i = q ( i ) and p | i = 1- q ( i ) for i = 1 , . . . , n . By the lecture this defines a sequential model. Let A 1 = { X 2 = 1 } and A = { X 2 = 0 } . A 1 and A represent the events that the chosen coin shows Head or Tail. As A = A c 1 it suffices to check P ( A 1 ) = 1 2 : P ( A 1 ) = ∑ ( i,j ) ∈ A 1 P (( i, j )) = n ∑ i =1 P (( i, 1)) = n ∑ i =1 p i p 1 | i = n ∑ i =1 1 n q ( i ) = 1 2 ....
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This note was uploaded on 07/07/2010 for the course MATH 170A 170A taught by Professor Richthammer during the Winter '10 term at UCLA.

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HW3 - Mathematics Department, UCLA T. Richthammer winter...

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