fall09finalsoln

# fall09finalsoln - University of Illinois Fall 2009 ECE 313...

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Unformatted text preview: University of Illinois Fall 2009 ECE 313: Final Exam Tuesday, December 15, 2009, 8:00 a.m. — 11:00 a.m. 100 Materials Science and Engineering Building 1. (a) A , B , and C are events such that 0 < P ( A ) < 1, 0 < P ( B ) < 1, and 0 < P ( C ) < 1. • P ( A ∪ B ) = P ( A c ∪ B c )- P ( A c )- P ( B c ) + 1 is a TRUE statement. Note that the right side is- P ( A c ∩ B c )+1 and the result follows from DeMorgan’s theorem. • P ( AB ) ≥ P ( A ) + P ( B )- 1 is a TRUE statement. Transposition gives P ( A ) + P ( B )- P ( AB ) = P ( A ∪ B ) ≤ 1 which obviously is true. • P ( A c | B ) P ( B ) + P ( A c | B c ) P ( B c ) = P ( A c ) is a TRUE statement. It is just the law of total probability applied to P ( A c ). • P ( A c | B ) P ( B ) + P ( A | B ) P ( B ) = P ( B ) is a TRUE statement. Note that P ( B ) is a common factor on the left side and P ( A c | B ) + P ( A | B ) = 1, • P ( A c | B ) P ( B ) + P ( A | B c ) P ( B c ) = P ( A ∪ B )- P ( AB ) is a TRUE statement. Both sides equal P ( A ⊕ B ). • P ( B | A ) = P ( A | B ) P ( A ) /P ( B ) is a FALSE statement. The similar-looking P ( B | A ) = P ( A | B ) P ( B ) /P ( A ) is, of course, just Bayes’ formula. • “If A and B are mutually exclusive events, then they are independent events” is a FALSE statement. Mutually exclusive events are independent only in the trivial case when at least one of the events has zero probability. • “If A , B , and C are independent events, then P ( ABC ) = P ( A ) P ( B ) P ( C )” is a TRUE statement. The condition P ( ABC ) = P ( A ) P ( B ) P ( C ) is one of four conditions that must hold for A , B , and C to be called independent events. • “If P ( ABC ) = P ( A ) P ( B ) P ( C ), then A , B , and C are independent ” is a FALSE statement. It is also necessary that P ( AB ) = P ( A ) P ( B ), P ( AC ) = P ( A ) P ( C ), and P ( BC ) = P ( B ) P ( C ) hold in order for A , B , and C to be independent events. (b) f X ( u ) is an even function and var ( X ) = 4. • F X ( u ) = F X (- u ) for all u,-∞ < u < ∞ is a FALSE statement. In fact, F X ( u ) = 1- F X (- u ) for all u,-∞ < u < ∞ . • E [ X 2 ] = 4 is a TRUE statement. E [ X 2 ] = var ( X ) + ( E [ X ]) 2 = var ( X ) = 4....
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fall09finalsoln - University of Illinois Fall 2009 ECE 313...

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