4. (10 points) Two dogs - Lisa and Cooper - share a population of N E N fleas. Fleas jump from one dog to another independently at rate A per minute. Let Xt denote the number of fleas on Lisa at time t (measured in minutes). We assume that (Xt)to is a birth-and-death process. Suppose there are x e {0, 1, ..., N} fleas on Lisa at time t = 0. (a) Compute the expected number mr(t) of fleas on Lisa at time t > 0, i.e., find mr(t) = E[Xt | Xo = x]. Hint: Use Kolmogorov's forward equation to show that the function mr(t) satisfies the linear ODE m'?(t) = -2\\mz(t) + N> with mr(0) = x. Then, recall that the solution to a linear ODE of the form f' (x) = a. f(x) +b, f(0) =c with constants a, b, ce R is given by f (20 ) = c+ - . ear a (b) Compute the limit lime . E[Xt | Xo = x], i.e., the expected number of fleas on Lisa in the long-run