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 = 1 n E [ S ( t )] = S (0) E E 1 n N ( t ) = n = S (0) X n =0 1 n P ( N ( t ) = n ) = S (0) X n =0 1 n e- t ( t ) n n ! = S (0) e- t X n =0 t n n ! = S (0) e- t e t = S (0) e t ( 1 - 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 2 (see bottom of pg. 282) = 2 2 n E [ S ( t ) 2 ] = S (0)...
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HW5Soln - IEOR 161 - Introduction to Stochastic Processes...

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