PROB Engineering Probability Class 1 Mon 2020-01-13

W Randolph Franklin (WRF), Rensselaer Polytechnic Institute (RPI)

2020-01-13 00:00

Source

Table of contents::

1 Topics

Syllabus and Intro.
Why probability is useful
1. AT&T installed bandwidth to provide level of iphone service (not all users want to use it simultaneously).
2. also web servers, roads, cashiers, ...
3. What is a fair price for a car or health or life insurance?
4. Will a pension plan go broke?
5. What would you pay today for the right to buy a share of Tesla (TSLA) on 6/30/20 for 400 dollars? (Today, 1/10/19, it is 478.) The answer is not simply $78. It is complicated because you don't have to buy if TSLA is below $400 then.
To model something
1. Real thing too expensive, dangerous, time-consuming (aircraft design).
2. Capture the relevant, ignore the rest.
3. Coin flip: relevant: it's fair? not relevant: copper, tin, zinc, ...
4. Validate model if possible.
Computer simulation model
1. For systems too complicated for a simple math equation (i.e., most systems outside school)
2. Often a graph of components linked together, e.g., with
  1. Matlab Simulink
  2. PSPICE
1. many examples, e.g. antilock brake, US economy
2. Can do experiments on it.
To make public policy: "Compas (Correctional Offender Management Profiling for Alternative Sanctions), is used throughout the U.S. to weigh up whether defendants awaiting trial or sentencing are at too much risk of reoffending to be released on bail." Slashdot.
Deterministic model
1. Resistor: V=IR
2. Limitations: perhaps not if I=1000000 amps. Why?
3. Limitations: perhaps not if I=0.00000000001 amps. Why?
Probability model
1. Roulette wheel: $p_i=\frac{1}{38}$ (ignoring http://www.amazon.com/Eudaemonic-Pie-Thomas-Bass/dp/0595142362 )
Terms
1. Random experiment: different outcomes each time it's run.
2. Outcome: one possible result of a random experiment.
3. Sample space: set of possible outcomes.
  1. Discrete, or
  2. Continuous.
1. Tree diagram of successive discrete experiments.
2. Event: subset of sample space.
3. Venn diagram: graphically shows relations.
Statistical regularity
1. $lim_{n\rightarrow\infty}f_k(n) =p_k$
2. law of large numbers
3. weird distributions (e.g., Cauchy) violate this, but that's probably beyond this course.
Properties of relative frequency
1. the frequencies of all the possibilities sum to 1.
2. if an event is composed of several outcomes that are disjoint, the event's probability is the sum of the outcomes' probabilities.
3. 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 $p_{pass}=0.9$ . (These numbers are fictitious.)
Axiomatic approach
1. Probability is between 0 and 1.
2. Probs sum to 1.
3. If the events are disjoint, then the probs add.
Building a model
1. Want to model telephone conversations where speaker talks 1/3 of time.
2. Could use an urn with 2 black, 1 white ball.
3. Computer random number generator easier.
Detailed example in more detail - phone system
1. Design telephone system for 48 simultaneous users.
2. Transmit packet of voice every 10msecs.
3. Only 1/3 users are active.
4. 48 channels wasteful.
5. Alloc only M<48 channels.
6. In the next 10msec block, A people talked.
7. If A>M, discard A-M packets.
8. How good is this?
9. n trials
10. $N_k(n)$ trials have k packets
11. frequency $f_k(n)=N_k(n)/n$
12. $f_k(n)\rightarrow p_k$ probability
13. We'll see the exact formula (Poisson) later.
14. average number of packets in one interval:
  
  $\frac{\sum_{k=1}^{48} kN_k(n)}{n} \rightarrow \sum_{k=1}^{48} kp_k = E[A]$
15. That is the expected value of A.
Probability application: unreliable communication channel.
1. Transmitter transmits 0 or 1.
2. Receiver receives 0 or 1.
3. However, a transmitted 0 is received as a 0 only 90% of the time, and
4. a transmitted 1 is received as a 1 only 80% of the time, so
5. if you receive a 0 what's the probability that a 0 was transmitted?
6. ditto 1.
7. (You don't have enough info to answer this; you need to know also the probability that a 0 was transmitted. Perhaps the transmitter always sends a 0.)
Another application: stocking spare parts:
1. There are 10 identical lights in the classroom ceiling.
2. The lifetime of each bulb follows a certain distribution. Perhaps it dies uniformly anytime between 1000 and 3000 hours.
3. As soon as a light dies, the janitor replaces it with a new one.
4. How many lights should the janitor stock so that there's a 90% chance that s/he won't run out within 5000 hours?