fall09final - University of Illinois Fall 2009 ECE 313...

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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. [60 points] (3 points per answer) Mark TRUE or FALSE for each statement below. No justification is required, but to discourage guessing, 3 points will be deducted for each incorrect answer (no penalty or gain for blank answers). A net negative score will reduce your total exam score. (a) A , B , and C are events such that 0 < P ( A ) < 1, 0 < P ( B ) < 1, and 0 < P ( C ) < 1. TRUE FALSE ± ± P ( A B ) = P ( A c B c ) - P ( A c ) - P ( B c ) + 1 . ± ± P ( AB ) P ( A ) + P ( B ) - 1 . ± ± P ( A c | B ) P ( B ) + P ( A c | B c ) P ( B c ) = P ( A c ) . ± ± P ( A c | B ) P ( B ) + P ( A | B ) P ( B ) = P ( B ) . ± ± P ( A c | B ) P ( B ) + P ( A | B c ) P ( B c ) = P ( A B ) - P ( AB ) . ± ± P ( B | A ) = P ( A | B ) P ( A ) /P ( B ). ± ± If A and B are mutually exclusive events, then they are independent events. ± ± If A , B , and C are independent events, then P ( ABC ) = P ( A ) P ( B ) P ( C ). ± ± If P ( ABC ) = P ( A ) P ( B ) P ( C ), then A , B , and C are independent events. (b) X is a continuous random variable whose probability density function f X ( u ) is an even function , that is, f X ( u ) = f X ( - u ) for all real numbers u . Let F X ( u ) denote the cumulative probability distribution function (CDF) of X , and assume that that var ( X ) = 4. TRUE FALSE ± ± F X ( u ) = F X ( - u ) for all u, -∞ < u < . ± ± E [ X 2 ] = 4. ± ± E [ | X | ] = 2. ± ± P { X > u } = F X ( - u ) for all u, - ∞ < u < . ± ± P
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fall09final - University of Illinois Fall 2009 ECE 313...

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