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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

Page1 / 6

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online