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Engineering Probability Class 5 Thu 2020-01-30

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1   Probability in the real world - enrichment

See examples in the random section below.

2   Homework 3

is online here , due in a week.

3   Chapter 2 ctd

  1. Example 2.28, p51. Chip quality control.
    1. Each chip is either good or bad.
    2. P[good]=(1-p), P[bad]=p.
    3. If the chip is good: P[still alive at t] = \(e^{-at}\)
    4. If the chip is bad: P[still alive at t] = \(e^{-1000at}\)
    5. What's the probability that a random chip is still alive at t?
  2. 2.4.1, p52. Bayes' rule. This lets you invert the conditional probabilities.
    1. \(B_j\) partition S. That means that
      1. If \(i\ne j\) then \(B_i\cap B_j=\emptyset\) and
      2. \(\bigcup_i B_i = S\)
    2. \(P[B_j|A] = \frac{B_j\cap A}{P[A]}\) \(= \frac{P[A|B_j] P[B_j]}{\sum_k P[A|B_k] P[B_k]}\)
    3. application:
      1. We have a priori probs \(P[B_j]\)
      2. Event A occurs. Knowing that A has happened gives us info that changes the probs.
      3. Compute a posteriori probs \(P[B_j|A]\)
  3. In the above diagram, what's the probability that an undergrad is an engineer?
  4. Example 2.29 comm channel: If receiver sees 1, which input was more probable? (You hope the answer is 1.)
  5. Example 2.30 chip quality control: For example 2.28, how long do we have to burn in chips so that the survivors have a 99% probability of being good? p=0.1, a=1/20000.
  6. Example: False positives in a medical test
    1. T = test for disease was positive; T' = .. negative
    2. D = you have disease; D' = .. don't ..
    3. P[T|D] = .99, P[T' | D'] = .95, P[D] = 0.001
    4. P[D' | T] (false positive) = 0.98 !!!

4   Bayes theorem ctd

  1. Wikipedia on Bayes theorem.

    We'll do the examples.

  2. We'll do these examples from Leon-Garcia in class.

  3. Example 2.28, page 51. I'll use e=0.1.

    Variant: Assume that P[A0]=.9. Redo the example.

  4. Example 2.30, page 53, chip quality control: For example 2.28, how long do we have to burn in chips so that the survivors have a 99% probability of being good? p=0.1, a=1/20000.

  5. Event A is that a random person has a lycanthopy gene. Assume P(A) = .01.

    Genes-R-Us has a DNA test for this. B is the event of a positive test. There are false positives and false negatives each w.p. (with probability) 0.1. That is, P(B|A') = P(B' | A) = 0.1

    1. What's P(A')?
    2. What's P(A and B)?
    3. What's P(A' and B)?
    4. What's P(B)?
    5. You test positive. What's the probability you're really positive, P(A|B)?

5   Chapter 2 ctd: Independent events

  1. 2.5 Independent events

    1. \(P[A\cap B] = P[A] P[B]\)
    2. P[A|B] = P[A], P[B|A] = P[B]

  2. A,B independent means that knowing A doesn't help you with B.

  3. Mutually exclusive events w.p.>0 must be dependent.

  4. Example 2.33, page 56.

    /images/fig214.jpg

  5. More that 2 events:

    1. N events are independent iff the occurrence of no combo of the events affects another event.
    2. Each pair is independent.
    3. Also need \(P[A\cap B\cap C] = P[A] P[B] P[C]\)
    4. This is not intuitive A, B, and C might be pairwise independent, but, as a group of 3, are dependent.
    5. See example 2.32, page 55. A: x>1/2. B: y>1/2. C: x>y

  6. Common application: independence of experiments in a sequence.

  7. Example 2.34: coin tosses are assumed to be independent of each other.

    P[HHT] = P[1st coin is H] P[2nd is H] P[3rd is T].

  8. Example 2.35, page 58. System reliability

    1. Controller and 3 peripherals.
    2. System is up iff controller and at least 2 peripherals are up.
    3. Add a 2nd controller.

  9. 2.6 p59 Sequential experiments: maybe independent

  10. 2.6.1 Sequences of independent experiments

    1. Example 2.36

  11. 2.6.2 Binomial probability

    1. Bernoulli trial flip a possibly unfair coin once. p is probability of head.
    2. (Bernoulli did stats, econ, physics, ... in 18th century.)

  12. Example 2.37

    1. P[TTH] = \((1-p)^2 p\)
    2. P[1 head] = \(3 (1-p)^2 p\)

  13. Probability of exactly k successes = \(p_n(k) = {n \choose k} p^k (1-p)^{n-k}\)

  14. \(\sum_{k=0}^n p_n(k) = 1\)

  15. Example 2.38

  16. Can avoid computing n! by computing \(p_n(k)\) recursively, or by using approximation. Also, in C++, using double instead of float helps. (Almost always you should use double instead of float. It's the same speed.)

  17. Example 2.39

  18. Example 2.40 Error correction coding