Fall 2010 HW07 - University of Illinois Fall 2010 ECE 313:...

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Unformatted text preview: University of Illinois Fall 2010 ECE 313: Problem Set 7: Solutions Continuous-type random variables, uniform and exponential distributions, and Poisson processes 1. [Continuous-type random variables I] (a) A = C = 0 ,B = 1 . (b) F ( c ) = , c ≤ c 2 / 2 , ≤ c < 1- c 2 / 2 + 2 c- 1 , 1 ≤ c < 2 1 , c ≥ 2 2. [Continuous-type random variables II] If E[X]=5/8, find a and b. Z 1 ( a + bu 2 ) du = 1 → a + b 3 = 1 Z 1 u ( a + bu 2 ) du = 3 / 5 , → a/ 2 + b/ 4 = 5 / 8 . Hence a=1/2, b=3/2. 3. [Uniform and exponential distribution I] (a) E [ | X- a | ] = Z a ( a- u ) du/L + Z L a ( u- a ) du/L = L 2 + a 2 L- a. d da () = 2 a L- 1 = 0 → a = L/ 2 Or we could rearrange the expression for E [ X ] to get E [ X ] = L 4 + ( a- L/ 2) 2 L , which by inspection is minimized at a = L 2 . (b) E [ | X- a | ] = Z a ( a- u ) λe- λu du + Z ∞ a ( u- a ) λe- λu du = a (1- e- λa ) + ae- λa + e- λa λ- 1 λ + ae- λa + e- λa λ- ae- λa = a- 1 λ + 2 e- λa λ Differentiation yields the minimum at...
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This note was uploaded on 02/09/2011 for the course ECE 313 taught by Professor Milenkovic,o during the Fall '08 term at University of Illinois, Urbana Champaign.

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Fall 2010 HW07 - University of Illinois Fall 2010 ECE 313:...

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