EECS 501__________________________PROBLEM SET #9__________________________Fall 2001

ASSIGNED: November 16, 2001. Read Stark and Woods pp. 371-407 on cont-time rps. SKIP Chap. 9.
DUE DATE: November 30, 2001. THIS WEEK: Cont-time rps and Bernoulli and Poisson processes.
  1. Stark and Woods #8.7. Compute KN, not RN. HINT: Use problem #3b of Problem Set #5.
  2. Stark and Woods #8.11. Characterizing a Markov process from its transition density function.
  3. Stark and Woods #8.15. Find Ky, not Ry. Compute fy(3),y(7),y(9)(Y3,Y7,Y9) if y(t) is Gaussian.
    1. A problem from a previous third exam: ("II"="independent increments")
    2. A zero-mean Gaussian II x(t) has Var[x(4)]=12. Compute fx(3),x(7),x(9)(X3,X7,X9).
    3. PROVE that a Markov process with fx(t)|x(s)(Xt|Xs)=fx(t)-x(s)(Xt-Xs) has II.
      1. A zero-mean iid u(n) with Var[u(n)]=3 is input into x(n)=½ x(n-1)+u(n) for n > 0.
      2. If x(0)=0, compute Kx(m,n). Show that x(n) "looks" WSS for large times n.
      3. Now let x(0) be a zero-mean random variable uncorrelated with u(n).
        For what value of Var[x(0)] does x(n) "look" WSS for ALL times n > 0?
        RECALL: total system response = zero-state response + zero-input response
        (due to input AND initial cond)=(due to input only)+(due to initial condition).

  4. We observe K arrivals in time T for a Poisson process with average arrival rate either 2 or 4.
    We have a priori information that "2" and "4" are equally likely, so we use a MAP rule.
    Compute the a posteriori probability Pr["2"|K arrivals in time T]. HINT: Bayes's rule.
  5. Two Geiger counters monitor radioactive particles emitted by two Uranium ore samples.
    Radioactivity is Poisson with average arrival rates 5/sec for sample #1 and 8/sec for sample #2.
    The Geiger counters are defective; counter #1 works only 80% of the time; counter #2 only 75%.
    Radioactive particle emissions and Geiger counter failures are independent of each other.
      When a particle is emitted and its counter works, the counter makes a click.
    1. Compute Pr[3 of next 5 clicks come from counter #1 (which measures sample #1 only)].
    2. Compute Pr[10 emissions in next 3 seconds, 7 from sample #1 and 3 from sample #2].
      Note that this asks for "emissions" from the ore, NOT "clicks" from the counters!
    3. Given that next emission is NOT detected, what is probability it came from sample #1?
    4. What is pmf for #emissions (from either sample) until counter #2 fails for the 7th time?
    5. Compute Pr[counter #1 fails 2 or more times in next 2 seconds] in 2 different ways.