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Unformatted text preview: Utah State University ECE 6010 Stochastic Processes Homework # 2 Solutions 1. Suppose X is a r.v. with c.d.f. F X . Prove the following: (a) F X is nondecreasing. Let b > a . F X ( b ) F X ( a ) = P ( X b ) P ( X a ) = P ( X a ) + P ( a < X b ) P ( X a ) = P ( a < X b ) . So F X ( b ) F X ( a ) for b > a , which means F X is nondecreasing. (b) lim a F X ( a ) = 1. lim a F X ( a ) = lim a P ( X a ) = P ( { : X ( ) a } ) = P () = 1 . (c) lim a  F X ( a ) = 0. lim a  F X ( a ) = lim a  P ( { : X ( ) a } ) = P ( ) = 1 . (d) F X is right continuous. Let B n = { : X ( ) a + 1 /n } for n = 1 , 2 , . . . . Note that this is a nested sequence, B 1 B 2 . We have lim n F X ( a + 1 /n ) = lim n P ( B n ) = P ( lim n B n ) by continuity of probability. But lim n B n = { : X ( ) a } , so lim n F X ( a + 1 /n ) = P ( X a ) . Since the limit from the right is equal to the limiting value, we have right conti nuity. (e) P ( a < X b ) = F X ( b ) F X ( a ) if b > a . P ( a < X b ) = P ( X b ) P ( X a ) = F X ( b ) F X ( a ). 1 (f) P ( X = a ) = F X ( a ) lim b a F X ( b ) . P ( X = a ) = P ( X a ) P ( X < a ) = F X ( a ) lim b a F X ( b ). Also, find expressions for P ( a X b ), P ( a X < b ) and P ( a < X < b ) in terms of F X ....
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 Spring '08
 Stites,M

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