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.

- Stark and Woods #8.7. Compute K
_{N}, not R_{N}. HINT: Use problem #3b of Problem Set #5.
- Stark and Woods #8.11. Characterizing a Markov process from its transition density function.
- Stark and Woods #8.15. Find K
_{y}, not R_{y}.
Compute f_{y(3),y(7),y(9)}(Y3,Y7,Y9) if y(t) is Gaussian.

A problem from a previous third exam: ("II"="independent increments")

- A zero-mean Gaussian II x(t) has Var[x(4)]=12. Compute f
_{x(3),x(7),x(9)}(X3,X7,X9).

- PROVE that a Markov process with f
_{x(t)|x(s)}(Xt|Xs)=f_{x(t)-x(s)}(Xt-Xs) has II.

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.
- If x(0)=0, compute K
_{x}(m,n). Show that x(n) "looks" WSS for large times n.
- 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).

- 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.

- 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.

- Compute Pr[3 of next 5 clicks come from counter #1 (which measures sample #1 only)].
- 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!
- Given that next emission is NOT detected, what is probability it came from sample #1?
- What is pmf for #emissions (from either sample) until counter #2 fails for the 7th time?
- Compute Pr[counter #1 fails 2 or more times in next 2 seconds] in 2 different ways.