hw6sol - Math 361 X1 Homework 6 Solutions Spring 2003...

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Math 361 X1 Homework 6 Solutions Spring 2003 Graded problems: 2(a);4(a)(b);5(b);6(iii); each worth 3 pts., maximal score is 12 pts. Problem 1. [3.1:4] Let X 1 and X 2 be the numbers obtained on two rolls of a fair die. Let Y 1 = max( X 1 ,X 2 ) and Y 2 = min( X 1 ,X 2 ). Find the joint distribution of (a) ( X 1 ,X 2 ), (b) ( Y 1 ,Y 2 ). Solution. (a) Values: The values of X 1 and X 2 are 1 ,... , 6, so the joint distribution is a 6 × 6 matrix with entries indexed by 1 , 2 ,... , 6. Computation of probabilities P ( k,l ) = P ( X 1 = k,X 2 = l ) : This is easy: P (1 , 1) = P (1 appears in first roll and 1 appears in second roll) = ± 1 6 ² 2 = 1 36 , and the same calculation gives P ( k,l ) = 1 36 ( k,l = 1 , 2 ,... , 6) Check: Since there are 36 entries and each is 1 / 36, the sum of all entries is 1. (b) Values: The possible values of Y 1 and Y 2 are again 1 ,... , 6, so the joint distribution is a 6 × 6 matrix with entries indexed by 1 , 2 ,... , 6. Computation of probabilities P ( k,l ) = P ( Y 1 = k,Y 2 = l ) : P (1 , 1) = P ( Y 1 = 1 ,Y 2 = 1) = P (max( X 1 ,X 2 ) = 1 , min( X 1 ,X 2 ) = 1) = P ( X 1 = 1 ,X 2 = 1) = 1 36 , P (2 , 1) = P ( Y 1 = 2 ,Y 2 = 1) = P (max( X 1 ,X 2 ) = 2 , min( X 1 ,X 2 ) = 1) = P ( X 1 = 1 ,X 2 = 2) + P ( X 1 = 2 ,X 2 = 1) = 1 36 + 1 36 = 1 18 , P (1 , 2) = P ( Y 1 = 1 ,Y 2 = 2) = P (max( X 1 ,X 2 ) = 1 , min( X 1 ,X 2 ) = 2) = 0 , and in general, P ( k,l ) = P ( Y 1 = k,Y 2 = l ) = 1 / 36 if l = k 1 / 18 if l < k 0 if l > k Check: The sum of these entries is 6 · (1 / 36) + 15 · (1 / 18) = 1. 1
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Problem 2. [3.1:6] A fair coin is tossed three times. Let X be the number of heads on the first two tosses, Y the number of heads on the last two tosses. (a) Find the joint distribution of X and Y . (b) Are X and Y independent? Justify your answer rigorously. (c) Find the distribution of X + Y . Solution. (a) Values: The values of X and Y are 0 , 1 , 2, so the joint distribution table is a 3 × 3 matrix with entries indexed by 0 , 1 , 2. Computation of probabilities P ( k,l ) = P ( X = k,y = l ) : Each of these 9 probability computa- tions reduces to the computation of the probabilities of the specific H/T sequences that correspond to given values of X and Y . For example, P (0 , 0) = P ( X = 0 ,Y = 0) = P (0 heads in tosses 1 and 2, 0 heads in tosses 2 and 3) = P (0 heads in tosses 1,2,3) = P ( TTT ) = ± 1 2 ² 3 = 1 8 , P (0 , 1) =
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This note was uploaded on 04/03/2008 for the course STAT 134 taught by Professor Aldous during the Spring '03 term at University of California, Berkeley.

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hw6sol - Math 361 X1 Homework 6 Solutions Spring 2003...

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