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zz-22 - Utah State University ECE 6010 Stochastic Processes...

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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 ) 0 . 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
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(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 . P ( a X b ) = P ( a < X b ) + P ( X = a ) = F X ( b ) - F X ( a ) + ( F X ( a ) - lim b a - F X ( b )) . P ( a X < b ) = P ( a < X b ) + P ( X = a ) - P ( X = b ) = F X ( b ) - F X ( a ) + ( F X ( a ) - lim c a - F X ( c )) - ( F X ( b ) - lim c b - F X ( c )) 2. Show that the following are valid p.m.f.s: (a) Binomial: f X ( a ) = n ! / (( n - a )! a !) π a (1 - π ) n - a if a ∈ { 0 , 1 , . . . , n } .
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