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# hw5 - Homework 5 On-campus students Turn in to TA during...

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Homework 5 On-campus students: Turn in to TA during class time on Wed., Nov. 9. Off-campus students: Turn in all problems by 11:55 PM on Wed., Nov. 16. SS-1. Show that E [ X ] = for the random variable with distribution function F X ( x ) = ( 1 - x - 1 ) u ( x - 1 ) . 1. Let g ( X ) = ba X , where a and b are positive constants and X is a Poisson random variable with mean α . Find E [ g ( X )] . SS-2. Problem 4.11 in Stark and Woods (see problems at end of PDF) Ans: 5 . 8 SS-3. Problem 4.12 in Stark and Woods (see problems at end of PDF) Ans: π SS-4. Problem 4.23 in Stark and Woods (see problems at end of PDF) 2. Consider the relation between E [ f ( X )] and f ( E [ X ]) . Are they equal? (a) First conduct the following experiment. Generate 100,000 random variables U that are distributed uniformly on [ 0 , 1 ) . Create new variables X = U and Y = U 2 . Estimate E [ U ] , E [ X ] , and E [ Y ] using the sample mean. Compare with E [ X ] with E [ U ] and E [ Y ] with ( E [ U ]) 2 . (b) Find E [ X ] and E [ Y ] analytically and compare with your answers to part (a). 3. Let X be a Gaussian random variable with mean μ and variance σ 2 . Determine E [( 2 X + 3 ) 2 ] in terms of μ and σ 2 . SS-5. (Kay, Prob. 11.25) Determine the mean and variance for the indicator random variable I A ( X ) as a function of P [ A ] , where I A ( x ) = 1 , x A 0 , x / A 4. (Modified from Kay, Prob. 11.26) Let the input to a rectifier circuit be a Gaussian random variable with mean zero and variance σ 2 . Determine the expected output power (if the output is Y , determine the expected output power as E [ Y 2 ] ) for the following two scenarios. (a) The rectifier is a half-wave rectifier that passes positive voltages undisturbed but outputs zero when the input voltage is negative. (b) The rectifier is a full-wave rectifier that outputs the absolute value of the input voltage.

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5. (Kay, Prob. 11.32) Provide a counterexample to disprove that Var [ g 1 ( X )+ g 2 ( X )] = Var [ g 1 ( X )]+ Var [ g 2 ( X )] .
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