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

# Lecture 8

- iclicker: Could you please reregister yourselves (with your RCSID). For most people, I just see the serial number. Thanks.
- Midterm exam in class next Tues.
- closed book.
- bring one 2-sided crib sheet. Either handwritten or printed is ok.
- We will schedule a review session on Monday evening, and extra office hours.

- 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. - Let's do some exercises from the book.
- 2.83 on page 90.
- 2.99.
- 3.5 on page 130.
- 3.9.
- 3.15.
- 3.26.

- 3.4.2 Conditional expected value
- {$ m_{X|B} = \sum x_k p_X(x|B) $}
- {$ E[X] = \sum E[X|B_i] P[B_i] $}
- {$ E[X^2] = \sum E[X^2|B_i] P[B_i] $}
- {$ VAR[X] \ne \sum VAR[X|B_i] P[B_i] $}

- conditional variance: {$ VAR[X|B] = E[X^2|B] - E[X|B]^2 $}
- Example 3.26.
- 3.5 More important discrete r.v
- Table 3.1: We haven't seen {$ G_X(z) $} yet.
- 3.5.4 Poisson r.v.
- The experiment is observing how many of a large number of rare events happen in, say, 1 minute.
- E.g., how many cosmic particles hit your DRAM, how many people to call center.
- The individual events are independent.
- The r.v. is the number that happen in that period.
- There is one parameter, {$\alpha$}. Often this is called {$\lambda$}.
- {$$ p(k) = \frac{\alpha^k}{k!}e^{-\alpha} $$}
- Mean and std dev are both {$\alpha$}.

- Example 3.30 p 122 call center queries; Prob of over 4 calls in 10 seconds.
- Example 3.31 packet arrivals: Prob of no packets in a particular interval.
- Poisson approximates binomial for fixed np as {$n\rightarrow\infty $}
- Example 3.32 Optical transmission errors.
- Zipf r.v.
- {$ p(k) = c/k $} where c is chosen to make {$ \sum_{k=1}^L p(k)=1 $}
- L is important because summing to {$ \infty $} slowly diverges.
- The population of the i-th largest city is said to follow this.
- This may apply to other statistics, like the sales of the i-th most popular book on Amazon.
- I used weasel words above because there is some argument, which is beyond this course.
- Zipf has a
*long tail*because rare events, cumulatively, are more common than you'd think.

- zeta r.v. - generalizes Zipf by adding exponent.
- {$ p(k) = c/k^\alpha $} where c is chosen to make {$ \sum_{k=1}^L p(k)=1 $}
- Sum to infinity exists for {$ \alpha>1 $}
- Mean exists for {$ \alpha>2 $}, variance exists for {$ \alpha>3 $}
- This is an example of a perfectly well defined prob distribution whose moments may not exist.