Review (page 29).

1. Random experiment, e.g., voltage at wall.
2. Sample space S of all possible outcomes.
3. Types of sample spaces:
   1. Finite, e.g., which head of a coin is up
   2. Infinite discrete (countable), e.g., how many tosses until a head shows.
   3. Infinite continuous (uncountable), e.g., time until a uranium atom decays
   4. Combos are possible, but not in this course.
4. Events are subsets of S.
5. Event class F of interesting events. For discrete (finite or infinite) S, F can be all subsets.
6. For continuous, that doesn't work for technical reasons. So, can use countable numbers of unions, complements of intervals.
7. Probability is a real number assigned to each event.
8. There are rules for what are legal probabilities.
   1. E.g., >=0, <=1, for disjoint events, probabilities sum.

Pairs of random variables (Chapter 5, page 233).

1. joint behavior: pmf, pdf, cmf (pmf is for discrete, pdf for continuous)
2. summarize behavior: joint moments
3. independent?, correlation
4. conditional probabilities

1 experiment -> 2 random variables

1. Get 2 random variables from the one experiment
2. Random experiment: pick a file on your computer.
3. 2 random vars: # whole blocks used and fraction of last partial block.
4. Random experiment: pick a student in class
5. 2 random vars: height, weight
6. experiment may be defined broadly:
7. Random experiment: toss 2 coins
8. 2 random vars: face showing on 1st coin, face showing on 2nd coin.

pmf, cdf, marginal probabilities for 2 coins

1. do in class for fair coins
2. do for unfair coins, 1st: p(H)=.6, 2nd: p(H)=.2.
3. that was too easy because they really were 2 separate experiments.
4. iclicker, what is p[both coins are heads]?
   • A: .04
   • B: .12
   • C: .25
   • D: .5
   • E: .36

level, cost of textbooks

1. compute joint pmf, joint cdf, marginal cdf, marginal probabilities
2. iclicker, what is p[cost <=$100]?
   • A: 0
   • B: 1/2
   • C: 7/16
   • D: 4/5
   • E: 1
3. Find P[cost<=$100 and level > 2]]

Continuous random variables - point in square

1. find pdf, cdf, marginals
2. P[X>Y]?

Continuous random variables - point in triangle

1. Triangle (0,0), (1,0), (0,1) (variant of exercise 5.16)
2. experiment: pick a point uniformly distributed
3. random variables: X,Y
4. intuition: they're correlated since large X means likely small Y.
5. pdf: find scale factor
6. find cdf by integrating pdf
7. P[X>Y]?

Independence

1. joint pmf/pdf is product of marginal pmfs/pdfs
2. equivalently joint cdfs is product of marginal cdfs
3. Are random variables in 2 coin example independent?
4. ... textbook cost ...?
5. ... square ...?
6. ... triangle ...?

Section 5.6

1. Find mean, variance of 2 variables in each of 4 previous examples.
2. joint moment
3. central moment
4. covariance {$ COV(X,Y) \overset{\Delta}{=} E\left[(X-E[X])(Y-E[Y])\right] = E[XY] - E[X]E[Y] $}
5. independent -> COV=0; reverse not always true for nonlinear dependence.