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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 ( { ω : X ( ω ) ≤ a } ) = P (Ω) = 1 . (c) lim a →∞ F X ( a ) = 0. lim a →∞ F X ( a ) = lim a →∞ P ( { ω : X ( ω ) ≤ a } ) = P ( ∅ ) = 1 . (d) F X is right continuous. Let B n = { ω ∈ Ω : X ( ω ) ≤ 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 = { ω : X ( ω ) ≤ 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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This note was uploaded on 03/01/2012 for the course ECE 6010 taught by Professor Stites,m during the Spring '08 term at Utah State University.
 Spring '08
 Stites,M

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