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Unformatted text preview: STA3007 Applied probability (0809) Solution to Assignment 9 Chapter 5 Problem 6.2 The key observation is that, if θ is uniform on [0 , 2 π ), and Y is independent of θ and has an arbitrary distribution, then Y + θ (mod 2 π ) is uniform on [0 , 2 π ). The proof of this statement may be difficult, but it is clear by intuition. As the particles are distributed on the surface of a circular region according to a spatial Poisson process, the angle of the polar coordinates of each point is uniformly distributed on [0 , 2 π ). Therefore, at the end of the point movement process, the points are still Pois son distributed over the region. Problem 6.3 Without loss of generality, let a be a nonnegative integer. Let T be the time of system failure, then { T > t } if and only if { Z ( t ) < a } . By observing Pr( T > t ) = Pr( Z ( t ) < a ) = Pr( Z ( t ) < = a 1) , we have E( T ) = 1 λ ∞ X n =0 G ( n ) ( a 1) = 1 λ " 1 + ∞ X n =1 G ( n ) ( a 1) # = 1 λ " 1 + ∞ X n =1 a 1 X k =0...
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This note was uploaded on 02/16/2011 for the course ECON 1224 taught by Professor Afdsa during the Spring '11 term at CUHK.
 Spring '11
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