Discrete-time stochastic processes

B show that pr z t z z x t x x is equal

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Unformatted text preview: place transform is ∏t; t≥0. 3.9. EXERCISES 129 Exercise 3.7. a) Let the inter-renewal interval of a renewal process have a second order Erlang density, fX (x) = ∏2 x exp(−∏x). Evaluate the Laplace transform of m(t) = E [N (t)]. b) Use this to evaluate m(t) for t ≥ 0. Verify that your answer agrees with (3.9). c) Evaluate the slope of m(t) at t = 0 and explain why that slope is not surprising. d) View the renewals here as being the even numbered arrivals in a Poisson process of rate ∏. Sketch m(t) for the process here and show one half the expected number of arrivals for the Poisson process on the same sketch. Explain the difference between the two. Exercise 3.8. a) Let N (t) be the number of arrivals in the interval (0, t] for a Poisson process of rate ∏. Show that the probability that N (t) is even is [1 + exp(−2∏t)]/2. Hint: Look at the power series expansion of exp(−∏t) and that of exp(∏t), and look at the sum P of the two. Compare this with n even Pr {N (t) = n}. e e b) Let N (t) be the number of even numbered arrivals in (0, t]. Show that N (t) = N (t)/2 − Iodd (t)/2 where Iodd (t) is a random variable that is 1 if N (t) is odd and 0 otherwise. h i e c) Use parts a and b to find E N (t) . Note that this is m(t) for a renewal process with 2nd order Erlang inter-renewal intervals. Exercise 3.9. Use Wald’s equality to compute the expected number of trials of a Bernoulli process up to and including the kth success. Exercise 3.10. A gambler with an initial finite capital of d > 0 dollars starts to play a dollar slot machine. At each play, either his dollar is lost or is returned with some additional number of dollars. Let Xi be his change of capital on the ith play. Assume that {Xi ; i=1, 2, . . . } is a set of IID random variables taking on integer values {−1, 0, 1, . . . }. Assume that E [Xi ] < 0. The gambler plays until losing all his money (i.e., the initial d dollars plus subsequent winnings). a) Let J be the number of plays until the gambler loses all his money. Is the weak law of large numbers sufficient to argue that limn→1 Pr {J > n} = 0 (i.e., that J is a random variable) or is the strong law necessary? b) Find E [J ]. Exercise 3.11. Let {Xi ; i ≥ 1} be IID binary random variables with PX (0) = PX (1) = 1/2. Let J be a non negative integer valued random variable defined on the above sample P space of binary sequences and let SJ = J=1 Xi . Find the simplest example you can in i which J is not a stopping rule for {Xi ; i ≥ 1} and where E [X ] E [J ] 6= E [SJ ]. Exercise 3.12. Let J = min{n | Sn ≤B or Sn ≥A}, where A is a positive integer, B is a negative integer, and Sn = X1 + X2 + · · · + Xn . Assume that {Xi ; i≥1} is a set of zero mean IID rv’s that can take on only the set of values {−1, 0, +1}, each with positive probability. 130 CHAPTER 3. RENEWAL PROCESSES a) Is J a stopping rule? Why or why not? Hint: Part of this is to argue that J is finite with probability 1; you do not need to construct a proof of this, but try to argue why it must be true. b) What are the possible values of SJ ? c) Find an expression for E [SJ ] in terms of p, A, and B , where p = Pr {SJ ≥ A}. d) Find an expression for E [SJ ] from Wald’s equality. Use this to solve for p. Exercise 3.13. Let {N (t); t≥0} be a renewal counting process generalized to allow for inter-renewal intervals {Xi } of duration 0. Let each Xi have the PMF Pr {Xi = 0} = 1 − ≤ ; Pr {Xi = 1/≤} = ≤. a) Sketch a typical sample function of {N (t); t≥0}. Note that N (0) can be non-zero (i.e., N (0) is the number of zero interarrival times that occur before the first non-zero interarrival time). b) Evaluate E [N (t)] as a function of t. c) Sketch E [N (t)] /t as a function of t. £ § d) Evaluate E SN (t)+1 as a function of t (do this directly, and then use Wald’s equality as a check on your work). e) Sketch the lower bound E [N (t)] /t ≥ 1/E [X ] − 1/t on the same graph with part (c). £ § f ) Sketch E SN (t)+1 − t as a function of t and find the time average of this quantity. £ § £ § g) Evaluate E SN (t) as a function of t; verify that E SN (t) 6= E [X ] E [N (t)]. Exercise 3.14. Consider a miner trapped in a room that contains three doors. Door 1 leads him to freedom after two-day’s travel; door 2 returns him to his room after four-day’s travel; and door 3 returns him to his room after eight-day’s travel. Suppose each door is equally likely to be chosen whenever he is in the room, and let T denote the time it takes the miner to become free. a) Define a sequence of independent and identically distributed random variables X1 , X2 , . . . and a stopping rule J such that T= J X Xi . i=1 b) Use Wald’s equality to find E [T ]. hP i P J c) Compute E Xi | J =n and show that it is not equal to E [ n Xi ]. i=1 i=1 d) Use part (c) for a second derivation of E [T ]. 3.9. EXERCISES 131 Exercise 3.15. Consider a non-arithmetic renewal counting process {N (t); t > 0. For a given t and δ , denote Pr {N (t + δ ) − N (t) = i} by Pi . a) Show that P2 ≤ P1 FX (δ ). b) Sho...
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