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Engineering Probability Class 8 Mon 2021-02-22

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1 Leon-Garcia chapter 2 ctd

1.1 2.6.4 p63 Geometric probability law

  1. Geometric probability

    1. Repeat Bernoulli experiment until 1st success.

    2. Define outcome to be # trials until that happens.

    3. Define q=(1-p).

    4. \(p(m) = (1-p)^{m-1}p = q^{m-1}p\) (p has 2 different uses here).

    5. \(\sum_{m=1}^\infty p(m) =1\)

    6. Probability that more than K trials are required = \(q^K\).

  2. Example: probability that more than 10 tosses of a die are required to get a 6 = \(\left(\frac{5}{6}\right)^{10} = 0.16\)

  3. Example 2.43 p64: error control by retransmission. A sent over a noisy channel is checksummed so the receiver can tell if it got mangled, and then ask for retransmission. TCP/IP does this.

    Aside: This works better when the roundtrip time is reasonable. Using this when talking to Mars is challenging.

1.2 2.6.5 p64 Sequences, chains, of dependent experiments

  1. Sequences, chains, of dependent experiments

    1. This is an important topic, but mostly beyond this course.

    2. In many areas, there are a sequence of observations, and the probability of each observation depends on what you observed before.

    3. It relates to Markov chains.

    4. Motivation: speech and language recognition, translation, compression

    5. E.g., in English text, the probability of a u is higher if the previous char was q.

    6. The probability of a b may be higher if the previous char was u (than if it was x), but is lower if the previous two chars are qu.

    7. Need to look at probabilities of sequences, char by char.

    8. Same idea in speech recognition: phonemes follow phonemes...

    9. Same in language understanding: verb follows noun...

  2. Example 2.44, p64.

  3. Example 2.45, p66.

3 Review

  1. From a deck of cards, I draw a card, look at it, put it back and reshuffle. Then I do it again. What's the probability that exactly one of the 2 cards is a heart?

  2. From a deck of cards, I draw a card, look at it, put it back and reshuffle. I keep repeating this. What's the probability that the 2nd card is the 1st time I see hearts?

  3. Text problem 2.62 page 88.

  4. Problem 2.76.

  5. Problem 2.126 page 95.

4 Wikipedia

Wikipedia's articles on technical subjects can be excellent. In fact, they often have more detail than you want. Here are some that are relevant to this course. Read at least the first few paragraphs.

  1. https://en.wikipedia.org/wiki/Outcome_(probability)

  2. https://en.wikipedia.org/wiki/Random_variable

  3. https://en.wikipedia.org/wiki/Indicator_function

  4. https://en.wikipedia.org/wiki/Gambler%27s_fallacy

  5. https://en.wikipedia.org/wiki/Fat-tailed_distribution

  6. https://en.wikipedia.org/wiki/St._Petersburg_paradox

5 Two types of testing errors

  1. There's an event A, with probability P[A]=p.

  2. There's a dependent event, perhaps a test or a transmission, B.

  3. You know P[B|A] and P[B|A'].

  4. Wikipedia:

    1. https://en.wikipedia.org/wiki/Type_I_and_type_II_errors

    2. https://en.wikipedia.org/wiki/Sensitivity_and_specificity

  5. Terminology:

    1. Type I error, False negative.

    2. Type II error, false positive.

    3. Sensitivity, true positive proportion.

    4. Selectivity, true negative proportion.