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Unformatted text preview: CS/EE 147 HW 1: Probability Refresher Guru: Raga Assigned: 03/30/10 Due: 04/09/10, Raga’s mailbox 1 , 1pm We encourage you to discuss these problems with others, but you need to write up the actual solutions alone. At the top of your homework sheet, list all the people with whom you discussed. Crediting help from other classmates will not take away any credit from you. Start early and come to office hours with your questions! 1 Alternative definition of expectation [18 points] In this problem, we will investigate alternative ways to calculate the expectation and higher moments of a random variable. This dual view of expectation will be important later in the course. (a) Consider a non-negative, discrete, integer valued random variable, X . Traditionally, the expectation is defined as E [ X ] = ∞ X n =0 nPr ( X = n ) . Prove that we can also calculate E [ X ] using E [ X ] = ∞ X n =0 Pr ( X > n ) . (b) For a continuous, non-negative random variable having p.d.f. f ( y ) the expectation is typically defined as E [ Y ] = Z ∞ yf ( y ) dy. Prove that we can also calculate E [ Y ] using E [ Y ] = Z ∞ ¯ F ( y ) dy, where ¯ F ( y ) = Pr ( Y ≥ y ) . (c) The i-th moment of Y is typically defined as E [ Y i ] = Z ∞ y i f ( y ) dy. Extend the analysis in the previous two parts to provide an equation for the i-th moment of Y in terms of ¯ F ( y ) . (d) Do (b) and (c) hold for general random variables (ones that are not non-negative), why or why not?...
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This document was uploaded on 01/05/2012.
- Fall '09