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

# Lecture 2

- iClicker
*Confession: this is the first time I've used this. Please**correct my mistakes.*- Answer this question (won't be graded): 2+2=? 1, B:2, C:3.9999, D:4, E: whatever you want it to be.
- Register your iClicker, using your RCS ID. E.g., I would register
as
*W Randolph Franklin frankwr*

- Statistical regularity
- {$ lim_{n\rightarrow\infty}f_k(n) =p_k $}
- law of large numbers
- weird distributions (e.g., Cauchy) violate this, but that's probably beyond this course.

- Properties of relative frequency
- the frequencies of all the possibilities sum to 1.
- if an
*event*is composed of several*outcomes*that are disjoint, the event's probability is the sum of the outcomes' probabilities. - E.g., If the event is your passing this course and the relevant outcomes are
grades A, B, C, D, with probabilities .3, .3, .2, .1, then p
_{pass}=.9. (These numbers are fictitious.)

- Axiomatic approach
- Prob is between 0 and 1.
- Probs sum to 1.
- If disjoint, probs add.

- Building a model
- Want to model telephone conversations where speaker talks 1/3 of time.
- Could use an urn with 2 black, 1 white ball.
- Computer random number generator easier.

- Detailed example - phone system
- Design telephone system for 48 simultaneous users.
- Xmit packet of voice every 10msecs.
- Only 1/3 users are active.
- 48 channels wasteful.
- Alloc only M<48 channels.
- In the next 10msec block, A people talked.
- If A>M, discard A-M packets.
- How good is this?
- n trials
- N
_{k}(n) trials have k packets - frequency f
_{k}(n)=N_{k}(n)/n - f
_{k}(n)->p_{k}probability - We'll see the exact formula (Poisson) later.
- average number of packets in one interval: {$$\frac{\sum_{k=1}^{48} kN_k(n)}{n} \rightarrow \sum_{k=1}^{48} kp_k = E[A] $$}
- That is the
*expected value*of A.

- Example of unreliable channel
- Want to xmit a bit: 0, 1
- It arrives wrong with prob. (w.p.) e, say 0.001
- Idea: xmit each bit 3 times and vote.
- 000 -> 0
- 001 -> 0
- 011 -> 1

- 3 bits arrive correct w.p. (1-e)
^{3}= 0.997002999 - 1 error w.p. 3(1-e)
^{2}e = 0.002994 - 2 errors w.p. 3(1-e)e
^{2}= 0.000002997 - 3 errors w.p. e
^{3}= 0.000000001 - corrected bit is correct if 0 or 1 errors, w.p.
(1-e)
^{3}+3(1-e)^{2}e = 0.999996999 - We reduced prob of error by factor of 1000.
- Cost: triple the transmission plus a little logic HW.

- iclicker exercise based on that (not graded)
- e=0.1
- What is prob of no error in 3 bits: A 0.1 B 0.3 C 0.001 D 0.729 E 0.9

- Example of text compression
- Simple way: Use 5 bits for each letter: A=00000, B=00001
- In English, 'E' common, 'Q' rare
- Use fewer bits for E than Q.
- Morse code did this 170 years ago.
- E = .
- Q = _ _ . _

- Aside: An expert Morse coder is faster than texting.
- English can be compressed to about 1 bit per letter (with difficulty); 2 bits is easy.
- Aside: there is so much structure in English text, that if you add the bit strings for 2 different texts bit-by-bit, they can usually mostly be reconstructed.
- That's how cryptoanalysis works.

- Example of reliable system design
- Nuclear power plant fails if
- water leaks
- and operator asleep (a surprising number of disasters happen in the graveyard shift).
- and backup pump fails
- or was turned off for maintenance
- .....

- What's the probability of failure?
- Design a better system? Coal mining kills.
- The backups themselves can cause problems (and are almost impossible to test).

- Nuclear power plant fails if

# Chapter 2

- Specifying random expt
- experimental procedure
- set of measurements

- Random expt may have subexpts and sequences of expts
- Outcome or sample point {$\xi$}: like an atom
- not decomposable

- Sample space S: set of outcomes
- |S|:
- finite, e.g. {H,T}
- discrete = countable, e.g., 1,2,3,4,...
- uncountable, e.g., {$\Re$}

- Event
- collection of outcomes, subset of S
- what we're interested in.
- e.g., outcome is voltage, event is V>5.
- certain event: S
- null event: {$\emptyset$}
- elementary event: one discrete outcome

- Set theory
- Sets: S, A, B, ...
- Universal set: U
- elements or points: a, b, c
- {$ a\in S, a\notin S $}, {$A\subset B$}
- Venn diagram
- empty set: {} or {$\emptyset$}
- operations on sets: equality, union, intersection, complement, relative complement
- properties (axioms): commutative, associative, distributive
- theorems: de Morgan