# hw3(3) - Homework assignment 3 ∗ February 7 2006 1 Using...

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Unformatted text preview: Homework assignment 3 ∗ February 7, 2006 1. Using the method of characteristics, solve the following equation: ∂x ∂a + e- t ∂x ∂t = − dx. Here d is a positive constant and a and t are interpreted as age and time. The initial and boundary conditions (both are supposedly known) are: x ( a, 0) = x ( a ) , x (0 , t ) = b ( t ) 2. This problem explores some properties of a discrete-time random walk on the integers Z . Let λ be the probability to move right and μ = 1 − λ the probability to move left (so a move must be made at each instant of time). Let p n ( t ) be the probability that at time t the position of the random walker is n . Assume that the random walk starts at n = 0, so that p (0) = 1 and p i (0) = 0 for i negationslash = 0. First show that p n ( t + 1) = λp n- 1 ( t ) + μp n +1 ( t ) , n ∈ Z . Second, using the method of the generating function, determine the mean m ( t ) and variance σ 2 ( t ) of the position of the random walker at time t . Verify the plausibility of your results....
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