#---------------------------------------------------------
# File:   MIT18_05S22_in-class5-script.txt
# Author: Jeremy Orloff
#
# MIT OpenCourseWare: https://ocw.mit.edu
# 18.05 Introduction to Probability and Statistics
# Spring 2022
# For information about citing these materials or our Terms of Use, visit:
# https://ocw.mit.edu/terms.
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Class 5

Jerry
  Slides 1,2 (1 minute)
     Agenda: Variance of discrete RV start continuous RV

  Slide 3 Studio 2 comments   (4 minutes)
     Decide on the fly whether to work the example or suggest they do on their own.

Jen
  Slide 4 variance and standard deviation (4 minutes)
     Highlight it's defined as the expected value of something
     Discuss units

  Slide 5 CLICKER question, order by variance (3 minutes)
     Short discussion based on spread

  Slide 6 CLICKER question, Var(X) = 0 ... (3 minutes)
     Discussion: variance = sum of non-negative terms. If one is nonzero then so is variance.
     Avoid the urge to keep talking about this

  Slide 7 Variance example (paused slide) (6 minutes)
     Work example -- directly, use board
     Paused slide:
     For E(X^2) - E(X)^2 formula.
        --Work this quickly by adding to X^2 to the table
        --Do not try to prove it. 
        --Remind class: It's in the reading, proof is easy

  Slide 8 Clicker question & Table question (CQ: 1 minute, Table Q: 4 minutes:)
     Paused slide
     No discussion move right to table question
     TABLE question: compute variance --staff should help them
       Discussion -- scale matters,
       Don't do the computation 

  Slide 9 Algebra of variance (2 minutes)
     No proofs
     Board questions will provide examples

  Slide 10 BOARD questions: Var of bernoulli, binomial etc (Work 10 min. discussion 4 min.)
     Discussion
     They will have gotten these, so no need to go through solutions
     Mention that using linearity requires independence
       
  Slide 11  Continuous RVs intro (1 min)

Jerry
  Slide 12  Calculus warmup (3 min)
      Assume they have seen this --even if it's not fresh in their minds
      Do not dwell on it. 
      Assure them we won't do hard integrals (though we may ask Wolfram Alpha to do them for us)
      Thinking of integrals conceptually as sums is the key

  Slide 13  Continuous random variables: range, pdf, cdf (3 min)
      Emphasize f(x) is a density and f(x)dx is a probability

  Slide 14 Visualization (2 min)
      They will see pictures like these a lot --they should get good at understanding them.
      Probabilities are computed by integration

  Slide 15 Properties of CDF (3 min)
      Same as discrete
      Highlight F' = f

  Slide 16 BOARD questions --pdf f(x) = cx^2 on [0,2] (Work 9 min. Discuss 5 min.)
      Judge by how they do whether this requires much discussion.

  Slide 17 Table discussion questions P(X=a) --NOT WITH CLICKERS  (3 min)
      (2 pauses)
      Resist the urge to keep explaining

  Slide 18 Table discussion questions valid cdf --NOT WITH CLICKERS (3 min)
      Discussion

*** If we have time: ***
*** These slides will be repeated in class5b on the assumption that we won't get to them in class5.
  Slide 19 Exponential distr. review (1 min.)
      Just say this is the density. If is often used to model waiting times.

  Slide 20 BOARD question Finish class
      Discussion next time (maybe)