1 No class next Thurs 3/25

Attend Pres Jackson's spring town meeting, or sleep late, or go on a hike, or whatever you want.

Homework that was due then is postponed.

2 Chapter 4, ctd

1. Example 4.22, page 169.

2. Section 4.4.4 Gamma r.v.

   Setting its 2 parameters to various values gives useful special cases.

   E.g. m-Erlang, the sum of m exponential r.v.

3. Section 4.4.5, Beta r.v., p 173.

   Setting its 2 parameters to various values gives useful special cases.

4. Section 4.4.6 Cauchy r.v.

   has no moments.

5. Problem 4.17, p 217.

6. Problem 4.27, p 218.

7. Problem 4.39, p 219.

8. Problem 4.56a, p 220.

9. Problem 4.68, p 222.

3 Chapter 5, Two Random Variables

1. One experiment might produce two r.v. E.g.,

   1. Shoot an arrow; it lands at (x,y).

   2. Toss two dice.

   3. Measure the height and weight of people.

   4. Measure the voltage of a signal at several times.

2. The definitions for pmf, pdf and cdf are reasonable extensions of one r.v.

3. The math is messier.

4. The two r.v. may be *dependent* and *correlated*.

5. The *correlation coefficient*, $\rho$, is a dimensionless measure of linear dependence. $-1\le\rho\le1$.

6. $\rho$ may be 0 when the variables have a nonlinear dependent relation.

7. Integrating (or summing) out one variable gives a marginal distribution.

8. We'll do some simple examples:

   1. Toss two 4-sided dice.

   2. Toss two 4-sided ''loaded'' dice. The marginal pmfs are uniform.

   3. Pick a point uniformly in a square.

   4. Pick a point uniformly in a triangle. x and y are now dependent.

9. The big example is a 2 variable normal distribution.

   1. The pdf is messier.

   2. It looks elliptical unless $\rho$=0.

10. I finished the class with a high level overview of Chapter 5, w/o any math.

4 Comic

Conditional Risk