#---------------------------------------------------------
# File:   MIT18_05S22_in-class6-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 6 Continuous expectation, variance; quantiles; LoLN, CLT

Jen
  Slide 1
 
  Slide 2 Agenda and announcements (3 min)

  Slide 3 Expected value (2 min.)
     This is a quick review of the reading. No need for long explanations
     Main point is x*f(x)dx is value time prob.
    
  Slide 4 Var and std dev (2 min.)
     This is a quick review of the reading. No need for long explanations

  Slide 5 Properties (2 min.)
     This is a quick review of the reading. No need for long explanations

  Slide 6 BOARD question compute mean, var, std dev of average (work 10 min. discuss 5 min.)
    Discussion
    Solution. Do a-e. Tell them (f) is on the posted solutions.

  Slide 7  Quantiles (3 min.)
    Special ones: median, quartiles, deciles, percentiles
    Inverse of the cdf,
    R: qnorm, qexp etc

  Slide 8 Greatest median 1 (4 min.)
    Have pdf with solution figure ready

  Slide 9 Greatest median 1 (4 min.)
    This is a puzzle.
    Have pdf with solution figure ready
    Let them guess and try to explain.
    Check time: if after 3:15 just point them to solutions

Jerry
  Slide 10  Histograms (1 minute)
    Define types of histograms  -- saw this in studio

  Slide 11 Example equal bin width (2 min)
    Tell them this will be short since they are about to do a board question
    Key points: look the same, density is a true density function

  Slide 12 Example unequal bin width (2 min)
    Tell them this will be short since they are about to do a board question
    Key points: On slide

  Slide 13:  BOARD question: histograms (work 10 minutes, discuss 3 minutes)
    Let's get them through this so discussion is not needed
     
  Slide 14 LoLN (4 min.)
    We all know this intuitively
    Don't belabor this. It is in the reading
    The basic idea is simple: with high probability the sample average is close to the mean.
    The LoLN is a theorem that formalizes it. 18.05 doesn't do proofs, but it's nice to know there is one. It is in the optional appendix to the class 6 reading.

  Slide 15 CQ Desperation (6 min.)
    Paused slide
    This is the plot of the movie Rounders, starring Cambridge's own Matt Damon.
    If time is short, let's just do a show of hands.

    KEY POINT: in the long run the LoLN says you achieve the expected average
    You have to decide if this is a good thing or not.

  Slide 16: LoLN histogram (2 min.)
    Rule of thumb: LoLN ---> histogram matches the density 

  If extra time: Go through problem 3 in the problems
     Explain that p = pnorm(x) --: x = qnorm(p)
     i.e. p = probability, x = possible value, 
          F(x) = p = probability, quantile(p) = x = value
     Do part b imprecisely, just eyeballing areas.