HW7solns - May 11, 2007 HW7 Problem Solutions #8.3.2. Fix...

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May 11, 2007 HW7 Problem Solutions #8.3.2. Fix X (0) = i , and consider the formal derivation (because the switch in order of limit and sum is not rigoroulsy justi±ed) E ( 1 h E ( X ( t + h ) - X ( t ) | X ( t )= k )) ± k 0 P 0 k ( t ) ± m q km m = ± k 0 P 0 k ( t ) ² ( k + 1)( + a )+( k - 1)( ) - ( a + k ( μ + λ )) k ³ = ± k 0 P 0 k ( t ) { k ( λ - μ )+ a } =( λ - μ ) M ( t a Then the equation for the mean M ( t )i s M ± ( t )=( λ - μ ) M ( t a, M (0) = i leading immediately to the ODE solution M ( t i + at if λ = μ , and otherwise M ( t ie ( λ - μ ) t + a μ - λ (1 - e ( λ - μ ) t ) For the second moment M 2 ( t EX 2 ( t ), we again derive formally M ± 2 ( t ± k 0 P 0 k ( t ) ² ( a + λk )( k +1) 2 +( μk )( k - 1) 2 - ( a λ + μ ) k ) k 2 ³ =2 ( λ - μ ) M 2 ( t )+(2 a + λ + μ ) M ( t a Using the initial condition M 2 (0) = i 2 , we conclude M 2 ( t i 2 e 2( λ - μ ) t + a 2( μ - λ ) (1 - e - 2( λ - μ ) t )(2 a + λ + μ ) i - a μ - λ ) 2 a + λ + μ μ - λ ( e ( λ - μ ) t - e 2( λ - μ ) t a ( μ - λ ) 2 (2 a + λ + μ )(1 - e 2( λ - μ ) t ) #2. This problem is extra-credit. #3. The stationary probability vector is easily found to be π . 4 ,.
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HW7solns - May 11, 2007 HW7 Problem Solutions #8.3.2. Fix...

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