A type of bacteria cell divides at a constant rate over time. (That is, the probability

Chapter 3, Problem 3.203

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A type of bacteria cell divides at a constant rate over time. (That is, the probability that a cell divides in a small interval of time t is approximately t.) Given that a population starts out at time zero with k cells of this bacteria and that cell divisions are independent of one another, the size of the population at time t, Y (t), has the probability distribution P[Y (t) = n] = n 1 k 1 ekt 1 et nk , n = k, k + 1,.... a Find the expected value and variance of Y (t) in terms of and t. b If, for a type of bacteria cell, = .1 per second and the population starts out with two cells at time zero, find the expected value and variance of the population after five seconds.