pset11_soln

# pset11_soln - 6.042/18.062J Mathematics for Computer...

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Unformatted text preview: 6.042/18.062J Mathematics for Computer Science May 3, 2005 Srini Devadas and Eric Lehman Problem Set 11 Solutions Due: 5PM on Friday, May 6 This is a mini-problem set. The first problem reviews basic facts about expectation. The second and third are typical final exam questions. Problem 1. Answer the following questions about expectation. (a) There are several equivalent definitions of the expectation of a random variable. If R is a random variable over sample space S , then we can compute Ex ( R ) by summing over individual outcomes or by summing over values in the range of R . Write down these two equivalent definitions of Ex ( R ) . Solution. Ex ( R ) = R ( w ) Pr ( w ) = v · Pr ( R = v ) w ∈ S v ∈ Range( R ) (b) Give another expression for Ex ( R ) that holds when R is a natural-valued random variable. Solution. ∞ Ex ( R ) = Pr ( R > k ) k =0 (c) Give a simple expression for Ex ( R ) that is valid when R is an indicator random variable. Solution. Ex ( R ) = Pr ( R = 1) (d) The expectation of a random variable can often be computed from expectations of simpler random variables. Rewrite each of the expressions below in terms of Ex ( R ) and Ex ( S ) . Note any conditions...
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pset11_soln - 6.042/18.062J Mathematics for Computer...

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