precept3sol - Computer Science 340 Reasoning About...

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Computer Science 340 Reasoning About Computation Precept 3 Problem 1 a) Prove that if X 1 and X 2 are independent random variables then E [ X 1 · X 2 ] = E [ X 1 ] E [ X 2 ]. b) Does E [ X 1 · X 2 ] = E [ X 1 ] E [ X 2 ] imply that X 1 and X 2 are independent random vari- ables? Solution: a) Let D 1 Z be the domain of X 1 and similarly D 2 be the domain of X 2 . According to the definition of expectation we know that: E [ X 1 ] E [ X 2 ] = i D 1 i · Pr[ X 1 = i ] j D 2 j · Pr[ X 2 = j ] = i D 1 j D 2 i · j · Pr[ X 1 = i ] · Pr[ X 2 = j ] Since X 1 , X 2 are independent we can proceed: = i D 1 j D 2 i · j · Pr[ X 1 = i ] · Pr[ X 2 = j | X 1 = i ] = i D 1 j D 2 i · j · Pr[ X 1 = i X 2 = j ] = t D 1 D 2 t · ( i,j ) D 1 × D 2 | i · j = t Pr[ X 1 = i X 2 = j ] = t D 1 D 2 t · Pr[ X 1 · X 2 = t ] = E [ X 1 · X 2 ] b) Let us define random variables X 1 and X 2 as follows: ( X 1 , X 2 ) = ( - 1 , 0) , with probability 1 4 (1 , 0) , with probability 1 4 (0 , 1) , with probability 1 2 Then E [ X 1 ] = 1 4 · ( - 1) + 1 4 · 1 + 1 2 · 0 = 0
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E [ X 2 ] = 1 4 · 0 + 1 4 · 0 + 1 2 · 1 = 1 2 E [ X 1 · X 2 ] = 1 4 · 0 + 1 4 · 0 + 1 2 · 0 = 0 Thus E [ X 1 · X 2 ] = E [ X 1 ] · E [ X 2 ] On the other hand, Pr( X 1 = 0) = 1 / 2, but Pr( X 1 = 0 | X 2 = 1) = 1, so X 1 and X 2 are not independent. Problem 2 Suppose you write the following C code to compute the maximum number stored in an array of size n : max = a[0]; for (int i = 1; i < n; i++) { if (max < a[i]) { max = a[i]; } } Assume that the array a is a permutation of the integers { 1 , 2 , . . . , n } and each per- mutation is equally likely. What is the expected number of times the line max = a[i]; is executed?
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