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Fall 2010 HW07

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

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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 ) = 0 , c 0 c 2 / 2 , 0 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 0 ( a + bu 2 ) du = 1 a + b 3 = 1 Z 1 0 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 0 ( 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 0 ( 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 a * = ln 2 λ . (Note: In general, the value of a that minimizes E [ | X - a | ] for a random variable X is the median of the distribution.)

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4. [Uniform and exponential distribution II] (a) Let B=radio comes from the first batch, L=lifetime of radio P { L c } = P ( L c | B ) P ( B ) + P ( L c | B c ) P ( B c ) P ( L c ) = 1 c 0 (1 - ( c/ 2) + e - c/ 10 )(1 / 2) 0 c
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• Fall '08
• Milenkovic,O
• Probability theory, Exponential distribution, Poisson process, 2 L, Memorylessness, Continuous-type random variables

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