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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 be the probability to move right and = 1 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) = p n- 1 ( t ) + p n +1 ( t ) , n Z . Second, using the method of the generating function, determine the mean m ( t ) and variance 2 ( t ) of the position of the random walker at time t . Verify the plausibility of your results....
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- Summer '06