#---------------------------------------------------------
# File:   MIT18_05S22_in-class7-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 7 Joint distributions, independence, covariance, correlation

There is a fair amount here. We'll need to be strict with time.

Jen
  Slide 1

  Slide 2  Announcements/Agenda for exam (3 minutes)
    There may be questions about the exam 
    BE BRIEF

  Slide 3  Joint distributions (1 minute)
  
  Slide 4-5 Joint pmf examples (1 for 4, 2 minutes for 5)
    For X,T show why p(4,2) = 0 and p(1,7) = 1/36

  Slide 6  Continuous joint distributions (2 minutes)
     This is conceptually just like one random variable
     Don't belabor

  Slide 7  Properties of pmf, pdf (2 minutes)
     Basically the same as before

  Slide 8  18.02 in 18.05 (3 minutes)
     We won't expect a lot.
     There may be some questions: BE BRIEF

  Slide 9  Discrete events (3 minutes)
     (paused slide) Solution on the second half.
     Walk through the white and orange squares in the first row.
     Use the board to write, e.g. Y=1, X=1, Y-X = 0 not >= 2. 
     Use words to say why this means the (1,1) cell is white.
     Don't have to write for each cell.

  Slide 10 Continuous events (2 minutes)
     (paused slide) Solution on the second half.

  Slide 11 Continuous CDF (2 minutes)

  Slide 12 Marginal distributions (5 minutes)
     Take enough time to do this for the example
     They will see it for continuous densities on the board problem

Jerry
  Slides 13-14 Board question (Work 12 minutes: discussion 6 minutes)
     Turn off full screen mode and try to show both slides
     Go through each part
     Highlight f(x,y) > 0, integral is 1
     Discuss F_X(x) = F(x,1)
     For (e) get the marginal pdf as the derivative of the cdf. Give it also as the integral of the joint pdf --but don't compute it out.

  Slide 15 Independence (1 minute)

  Slide 16-18 CQ Independence (5 minutes)
      IN CQ II point out that the 0's are giveaways that it is not a product.
      In CQ III choice iii is a bit of a joke. Many students won't see this as a product.
      These should be relatively quick. 
      The key point is that independence means the cells are the 
      products of the marginals

  Slide 19 Covariance definition (2 minutes)
      Note how just like variance it is defined as an expected value

 Slide 20 Properties. (2 minutes)
      Don't belabor
      Highlight 3 and 4

  Slide 21 Table question Cov = 0, not independent (4 minutes)
      (paused slide)
      Take the time to compute the covariance explicitly use E[XY] - E[X]E[Y]
      Copy the table to the board, so you can point to the cells as you multiply and add XYp(x,y)

  Slide 22  Correlation definition (4 minutes)

  Slide 23  Board Question (Work 10 minutes, Discussion 6 minutes)

-----------------------------------
Anything after this is a bonus

Slide 24 Correlation is not causation (5 minutes)
   Discussion is in the solutions
   Probably just choose 2 to discuss. One should be HRT and CHD.

Slide 25 Correlation is not causation (1 minute)
   This is the lesson of the previous slides

Slide 26 Sums of overlapping uniform
   SKIP THIS. For the plots it's enough to see how the correlation affects the spread
   This is in the reading as are the plots.

Slide 27 Scatter plots (2 minutes)
   Show as corr goes to 1, the scatter plot becomes thinner.
   The axes are centered on the means. Positive correlation means more of the scatter is in 1st and 3rd quadrants.

Slide 28 Intuiton check
   If by any chance we get here: note this is the studio for tomorrow
   Covariance stays at 1/4, correlation decreases as n increases.

Slide 29 Board question 
   Do the problem in slide 28.
   Won't get to this. 
   We'll simulate it studio on Friday