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Unformatted text preview: Exam 1 - Solution ORIE 3500/5500: Engineering Probability and Statistics II Question 1 (i) (a) The events A and B are independent if P ( A B ) = P ( A ) P ( B ). If P ( B ) > 0, this is equivalent to P ( A | B ) = P ( A ), i.e. whether or not B has happened, does not influence the likelihood of A . (b) CDF of a random variable X is a function F : R [0 , 1], such that F ( x ) = P ( X x ) . A CDF characterizes the distribution of all random variables, including both discrete and absolutely continuous random variables. (c) PDF of an absolutely continuous random variable Y is a function f : R [0 , ) such that Z - f ( t ) dt = 1 and P ( Y y ) = Z y- f ( t ) dt. (d) Expected value of a discrete random variable Z taking values z 1 ,z 2 ,...,z n with prob- abilities p 1 ,p 2 ,...,p n is E ( Z ) = n X i =1 z i p i . (ii) (a) Denote by the sample space, F is the family of events. A mapping P : F R is called a probability law, if the following holds: i. For any event A , P ( A ) 0. ii. P () = 1. iii. If A 1 ,A 2 ,... are disjoint events, then P ( A 1 A 2 ) = P ( A 1 ) + P ( A 2 ) + ....
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This homework help was uploaded on 02/20/2009 for the course ORIE 3500 taught by Professor Weber during the Fall '08 term at Cornell University (Engineering School).
- Fall '08