ECSE-2500, Engineering Probability, Spring 2010, Rensselaer Polytechnic Institute

Lecture 8

1. iclicker: Could you please reregister yourselves (with your RCSID). For most people, I just see the serial number. Thanks.
2. Midterm exam in class next Tues.
   1. closed book.
   2. bring one 2-sided crib sheet. Either handwritten or printed is ok.
   3. We will schedule a review session on Monday evening, and extra office hours.
3. Homework solutions are online. Username = ep. I'll give out the password in class. This is just to keep the answers away from the search engines.
4. Let's do some exercises from the book.
   1. 2.83 on page 90.
   2. 2.99.
   3. 3.5 on page 130.
   4. 3.9.
   5. 3.15.
   6. 3.26.
5. 3.4.2 Conditional expected value
   1. {$ m_{X|B} = \sum x_k p_X(x|B) $}
   2. {$ E[X] = \sum E[X|B_i] P[B_i] $}
   3. {$ E[X^2] = \sum E[X^2|B_i] P[B_i] $}
   4. {$ VAR[X] \ne \sum VAR[X|B_i] P[B_i] $}
6. conditional variance: {$ VAR[X|B] = E[X^2|B] - E[X|B]^2 $}
7. Example 3.26.
8. 3.5 More important discrete r.v
9. Table 3.1: We haven't seen {$ G_X(z) $} yet.
10. 3.5.4 Poisson r.v.
    1. The experiment is observing how many of a large number of rare events happen in, say, 1 minute.
    2. E.g., how many cosmic particles hit your DRAM, how many people to call center.
    3. The individual events are independent.
    4. The r.v. is the number that happen in that period.
    5. There is one parameter, {$\alpha$}. Often this is called {$\lambda$}.
    6. {$$ p(k) = \frac{\alpha^k}{k!}e^{-\alpha} $$}
    7. Mean and std dev are both {$\alpha$}.
11. Example 3.30 p 122 call center queries; Prob of over 4 calls in 10 seconds.
12. Example 3.31 packet arrivals: Prob of no packets in a particular interval.
13. Poisson approximates binomial for fixed np as {$n\rightarrow\infty $}
14. Example 3.32 Optical transmission errors.
15. Zipf r.v.
    1. {$ p(k) = c/k $} where c is chosen to make {$ \sum_{k=1}^L p(k)=1 $}
    2. L is important because summing to {$ \infty $} slowly diverges.
    3. The population of the i-th largest city is said to follow this.
    4. This may apply to other statistics, like the sales of the i-th most popular book on Amazon.
    5. I used weasel words above because there is some argument, which is beyond this course.
    6. Zipf has a long tail because rare events, cumulatively, are more common than you'd think.
16. zeta r.v. - generalizes Zipf by adding exponent.
    1. {$ p(k) = c/k^\alpha $} where c is chosen to make {$ \sum_{k=1}^L p(k)=1 $}
    2. Sum to infinity exists for {$ \alpha>1 $}
    3. Mean exists for {$ \alpha>2 $}, variance exists for {$ \alpha>3 $}
    4. This is an example of a perfectly well defined prob distribution whose moments may not exist.