Dr. Hackney STA Solutions pg 12

Dr. Hackney STA Solutions pg 12 - k ) + P ( A i A j ( k A k...

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Second Edition 1-9 1.41 a. P ( dash sent | dash rec) = P ( dash rec | dash sent) P ( dash sent) P ( dash rec | dash sent) P ( dash sent) + P ( dash rec | dot sent) P ( dot sent) = (2 / 3)(4 / 7) (2 / 3)(4 / 7) + (1 / 4)(3 / 7) = 32 / 41 . b. By a similar calculation as the one in (a) P (dot sent | dot rec) = 27 / 434. Then we have P ( dash sent | dot rec) = 16 43 . Given that dot-dot was received, the distribution of the four possibilities of what was sent are Event Probability dash-dash (16 / 43) 2 dash-dot (16 / 43)(27 / 43) dot-dash (27 / 43)(16 / 43) dot-dot (27 / 43) 2 1.43 a. For Boole’s Inequality, P ( n i =1 ) n X i =1 P ( A i ) - P 2 + P 3 + ··· ± P n n X i =1 P ( A i ) since P i P j if i j and therefore the terms - P 2 k + P 2 k +1 0 for k = 1 ,..., n - 1 2 when n is odd. When n is even the last term to consider is - P n 0. For Bonferroni’s Inequality apply the inclusion-exclusion identity to the A c i , and use the argument leading to (1.2.10). b. We illustrate the proof that the P i are increasing by showing that P 2 P 3 . The other arguments are similar. Write P 2 = X 1 i<j n P ( A i A j ) = n - 1 X i =1 n X j = i +1 P ( A i A j ) = n - 1 X i =1 n X j = i +1 " n X k =1 P ( A i A j A
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Unformatted text preview: k ) + P ( A i A j ( k A k ) c ) # Now to get to P 3 we drop terms from this last expression. That is n-1 X i =1 n X j = i +1 " n X k =1 P ( A i A j A k ) + P ( A i A j ( k A k ) c ) # n-1 X i =1 n X j = i +1 " n X k =1 P ( A i A j A k ) # n-2 X i =1 n-1 X j = i +1 n X k = j +1 P ( A i A j A k ) = X 1 i<j<k n P ( A i A j A k ) = P 3 . The sequence of bounds is improving because the bounds P 1 ,P 1-P 2 + P 3 ,P 1-P 2 + P 3-P 4 + P 5 ,... , are getting smaller since P i P j if i j and therefore the terms-P 2 k + P 2 k +1 0. The lower bounds P 1-P 2 ,P 1-P 2 + P 3-P 4 ,P 1-P 2 + P 3-P 4 + P 5-P 6 ,... , are getting bigger since P i P j if i j and therefore the terms P 2 k +1-P 2 k 0....
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This note was uploaded on 02/03/2012 for the course STA 1014 taught by Professor Dr.hackney during the Spring '12 term at UNF.

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