1 Link to lecture videos

is now in the top menu bar.

2 Midterm exam 1

Will be on Mon March 1. To accommodate students in China, the exam will be administered twice.

3 Fictional university, try 2

1. Consider a fictional university that has both undergrads and grads. It also has both Engineers and others:

   

2. Compute P[E|U], P[E|U'], P[U|E], etc.

4 Bayes theorem ctd

1. Wikipedia on Bayes theorem.

   We'll do the defective item example, using both numbers and Bayes rule.

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

3. Example 2.28, page 51. Assume P[A0]=1/3. P[e]=.1 What is P[A0|B0]?

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 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 2.32, page 55.

5. Example 2.33, page 56.

   

6. 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

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

8. 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].

9. 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.

10. 2.6 p59 Sequential experiments: maybe independent

11. 2.6.1 Sequences of independent experiments

    1. Example 2.36

12. 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.)

13. Example 2.37

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

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

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

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

16. Example 2.38

17. 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 on normal CPUs.)

18. Example 2.39

19. Example 2.40 Error correction coding