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131A_1_hw5 - if β 1 t = kβ t then 1-F 1 x =[1-F x k 1 10...

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EE 131A Problem Set #5 Probabilities Wednesday, November 9, 2005 Instructor: Vwani Roychowdhury Due: Wednesday, November 17, 2005 1. Problem 3.4 2. Problem 3.14 3. Problem 3.15 (b) 4. Problem 3.17 5. Problem 3.23 6. Problem 3.27 7. Problem 3.33 8. Problem 3.39 9. Optional Problem ( Solution Will be provided, but you need not submit it ) A system is put into operation at t = 0; its time of failure is a r.v. x with distribution F ( x ) and density f ( x ). We denote by β ( t ) dt the probability that system fails in the interval ( t, t + dt ), assuming that it did not fail up to time t . (a) Show that if β ( t ) = kt , then f ( x ) is a Rayleigh density. (b) Show that if x is uniformly distributed in the interval (0 , T ), then β ( t ) = 1 / ( T - t ) in this interval. (c) If β ( t ) = c =constant for 0 t T and zero for t > T , find f ( x ) (d) With β 1 ( t ), f 1 ( x ), F 1 ( x ) the corresponding quantities of another system, show that
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Unformatted text preview: if β 1 ( t ) = kβ ( t ), then 1-F 1 ( x ) = [1-F ( x )] k . 1 10. Optional Problem ( Solution Will be provided, but you need not submit it ) The space of an experiment consists of all points in the unit interval (0 , 1). All intervals are events with probability equal to their length: P [0 ≤ ζ ≤ y ] (1) We are given a continuous, monotone increasing function y = G ( x ), such that G (-∞ ) = 0, G ( ∞ ) = 1. With x = H ( y ) its inverse, we have G [ H ( y )] = y for any y in the interval (0,1). A r.v. x is so deFned that x ( ζ ) = H ( ζ ), where ζ is an outcome of the preceding experiment. Show that the distribution of x equals G ( x ). 2...
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