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Unformatted text preview: Virginia Tech The Bradley Department of Electrical and Computer Engineering ECE 5605 – Stochastic Signals and Systems – Fall 08 Problem Set 3 Problem 1. (a) Given that the events A 1 ,...,A n form a partition of S , and using the total probability theorem, show that F X ( x ) = F X ( x  A 1 ) P [ A 1 ] + ... + F X ( x  A n ) P [ A n ] and f X ( x ) = f X ( x  A 1 ) P [ A 1 ] + ... + f X ( x  A n ) P [ A n ] . (b) Using Bayes’ Theorem, show that P [ A  x 1 < X ≤ x 2 ] = F X ( x 2  A ) F X ( x 1  A ) F X ( x 2 ) F X ( x 1 ) P [ A ] . (1) (c) The conditional probability P [ A  X = x ] of the event A assuming X = x cannot be defined using Bayes’ Theorem because, in general, P [ X = x ] = 0. We shall define it as a limit P [ A  X = x ] = lim Δ x → P [ A  x < X ≤ x + Δ x ] . (2) With x 1 = x and x 2 = x + Δ x , we conclude from (1) and (2) that f X ( x  A ) = P ( A  X = x ) P ( A ) f X ( x ) . (3) (This is the continuous version of Bayes’ theorem.) Using (3), show that(This is the continuous version of Bayes’ theorem....
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 Fall '08
 DASILVER
 Probability theory

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