.. title: Engineering Probability Class 5 Thu 2020-01-30 .. slug: class05 .. date: 2020-01-29 .. tags: class .. link: .. description: .. type: text .. has_math: true .. sectnum:: .. contents:: Table of contents:: .. Probability in the real world - enrichment ------------------------------------------ See examples in the random section below. Homework 3 ---------- is online `here <../posts/homework03.html>`_ , due in a week. Chapter 2 ctd ------------- #. Example 2.28, p51. Chip quality control. a. Each chip is either *good* or *bad*. #. P[good]=(1-p), P[bad]=p. #. If the chip is good: P[still alive at t] = :math:`e^{-at}` #. If the chip is bad: P[still alive at t] = :math:`e^{-1000at}` #. What's the probability that a random chip is still alive at t? #. 2.4.1, p52. Bayes' rule. This lets you invert the conditional probabilities. a. :math:`B_j` partition S. That means that i. If :math:`i\ne j` then :math:`B_i\cap B_j=\emptyset` and #. :math:`\bigcup_i B_i = S` #. :math:`P[B_j|A] = \frac{B_j\cap A}{P[A]}` :math:`= \frac{P[A|B_j] P[B_j]}{\sum_k P[A|B_k] P[B_k]}` #. application: i. We have a priori probs :math:`P[B_j]` #. Event A occurs. Knowing that A has happened gives us info that changes the probs. #. Compute a posteriori probs :math:`P[B_j|A]` #. In the above diagram, what's the probability that an undergrad is an engineer? #. Example 2.29 comm channel: If receiver sees 1, which input was more probable? (You hope the answer is 1.) #. 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. #. Example: False positives in a medical test a. T = test for disease was positive; T' = .. negative #. D = you have disease; D' = .. don't .. #. P[T|D] = .99, P[T' | D'] = .95, P[D] = 0.001 #. P[D' | T] (false positive) = 0.98 !!! Bayes theorem ctd -------------------------- #. `Wikipedia on Bayes theorem `_. We'll do the examples. #. We'll do these examples from Leon-Garcia in class. #. Example 2.28, page 51. I'll use e=0.1. Variant: Assume that P[A0]=.9. Redo the example. #. 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. #. 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 #. What's P(A')? #. What's P(A and B)? #. What's P(A' and B)? #. What's P(B)? #. You test positive. What's the probability you're really positive, P(A|B)? Chapter 2 ctd: Independent events --------------------------------- #. 2.5 Independent events a. :math:`P[A\cap B] = P[A] P[B]` #. P[A|B] = P[A], P[B|A] = P[B] #. A,B independent means that knowing A doesn't help you with B. #. Mutually exclusive events w.p.>0 must be dependent. #. Example 2.33, page 56. .. image:: /images/fig214.jpg #. More that 2 events: a. N events are independent iff the occurrence of no combo of the events affects another event. #. Each pair is independent. #. Also need :math:`P[A\cap B\cap C] = P[A] P[B] P[C]` #. *This is not intuitive* A, B, and C might be pairwise independent, but, as a group of 3, are dependent. #. See example 2.32, page 55. A: x>1/2. B: y>1/2. C: x>y #. Common application: independence of experiments in a sequence. #. 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]. #. Example 2.35, page 58. System reliability a. Controller and 3 peripherals. #. System is up iff controller and at least 2 peripherals are up. #. Add a 2nd controller. #. 2.6 p59 Sequential experiments: *maybe* independent #. 2.6.1 Sequences of independent experiments a. Example 2.36 #. 2.6.2 Binomial probability a. *Bernoulli trial* flip a possibly unfair coin once. *p* is probability of head. #. (Bernoulli did stats, econ, physics, ... in 18th century.) #. Example 2.37 a. P[TTH] = :math:`(1-p)^2 p` #. P[1 head] = :math:`3 (1-p)^2 p` #. Probability of exactly k successes = :math:`p_n(k) = {n \choose k} p^k (1-p)^{n-k}` #. :math:`\sum_{k=0}^n p_n(k) = 1` #. Example 2.38 #. Can avoid computing n! by computing :math:`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.) #. Example 2.39 #. Example 2.40 Error correction coding Xkcd comic ---------- `Linear Regression `_