STA3007_0506_t10_ssol

STA3007_0506_t10_ssol - STA 3007 Applied Probability...

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STA 3007 Applied Probability 2005 Tutorial 10 Suggested Solution 1. The Uniform Distribution and Poisson Processes i. M = E [ X ( t ) k =1 2000 e - βW k ] = 2000 E [ X ( t ) k =1 e - βW k ] = 2000 n =1 E [ X ( t ) k =1 e - βW k | X ( t ) = n ] Pr { X ( t ) = n } E [ X ( t ) k =1 e - βW k | X ( t ) = n ] = E [ n k =1 e - βU k ] = nE [ e - βU i ] = n R t 0 e - βU 1 t du = - n ( ) - 1 R t 0 e - βU d ( - βu ) = n βt (1 - e - βt ) M = 2000 n =1 n βt (1 - e - βt ) Pr { X ( t ) = n } = 2000 βt (1 - e - βt ) n =1 nPr { X ( t ) = n } = 2000 βt (1 - e - βt ) E [ X ( t )] = 2000 λ β (1 - e - βt ) 2. Compound Poisson Processes i. X ( t ) : Number of customer in t-th day Poisson with λ = 10 Y k : Amount of money pay by the k-th customer exp with θ = 1 100 Z ( t ) = X ( t ) k =1 Y k E [ Z (10)] = λμt = (10)(10)(100) = 10000 var [ Z (10)] = λ ( ν 2 + μ 2 ) t = 10(100 2 + 100 2 )10 = 1 , 000 , 000 ii. E [ Y i ] = 1 θ , var [ Y i ] = 1 θ 2
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STA3007_0506_t10_ssol - STA 3007 Applied Probability...

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