HW5Soln

# HW5Soln - IEOR 161 Introduction to Stochastic Processes...

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Unformatted text preview: IEOR 161 - Introduction to Stochastic Processes Spring 2010 HW5 Solutions ** Note that the numbering is from Ross 9th Edition 5.36 a) E [ S ( t )] = E S (0) N ( t ) Y i =1 X i = S (0) · E E N ( t ) Y i =1 X i N ( t ) = n ⇒ E N ( t ) Y i =1 X i N ( t ) = n = E " n Y i =1 X i N ( t ) = n # = n Y i =1 E [ X i | N ( t ) = n ] = n Y i =1 1 μ = 1 μ n ⇒ E [ S ( t )] = S (0) · E E 1 μ n N ( t ) = n = S (0) ∞ X n =0 1 μ n · P ( N ( t ) = n ) = S (0) ∞ X n =0 1 μ n e- λt ( λt ) n n ! = S (0) e- λt ∞ X n =0 λt μ n n ! = S (0) e- λt e λt μ = S (0) e λt ( 1 μ- 1 ) 1 b) E [ S ( t ) 2 ] = E S (0) 2 N ( t ) Y i =1 X 2 i = S (0) 2 · E E N ( t ) Y i =1 X 2 i N ( t ) = n ⇒ E N ( t ) Y i =1 X 2 i N ( t ) = n = E " n Y i =1 X 2 i N ( t ) = n # = n Y i =1 E [ X 2 i | N ( t ) = n ] = n Y i =1 2 μ 2 (see bottom of pg. 282) = 2 μ 2 n ⇒ E [ S ( t ) 2 ] = S (0)...
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HW5Soln - IEOR 161 Introduction to Stochastic Processes...

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