# Conditional Distributionss

### Fall 2012

#### M / W / F, 12:05 - 12:55 pm

Instructor

Will Perkins

wperkins3@math.gatech.edu

Office: Skiles 017

Office Hours: Wednesday 10 am - 12pm, or by appointment

Topics
• Sections: 1.4, 3.2, 3.6, 3.7, 4.2, 4.5, 4.6
• Definitions:
1. Conditional Probability
2. Independent Random Variables
3. Joint Distribution Function
4. Joint probability mass function
5. Joint density function
6. Joint Distribution
7. Marginal distribution
8. Conditional Distribution Function
9. Conditonal PMF
10. Conditonal Density Function
11. Conditonal Expectation (of discrete and continuous RV's)
• Theorems: 3.2.3, 3.6.3, 3.7.4, 3.7.6, 4.6.5, 4.6.10
• Sample Questions:
1. Exercise 4.6.1
2. What is the conditional expectation of S_5 given that S_100 = 20?
3. If X is Poisson with mean 1, what is E(X | X > 1)?
4. What is E(S_n | S_n >0)?
5. Exercise 4.7.9
6. Let X,Y be two jointly continuous RV's (what does this mean precisely?). Does it make sense to ask for the conditonal expectation of Y given that X=1? Why or why not?
7. Let X,Z be iid standard normals, and Y = X + Z. What is E(Y|X)? What is E(Y^2|Z)?
8. If X and Y have Poisson distributions with mean 10, what is the distribution of X conditioned on X+Y?
9. How would you define the conditional variance of Y given X? Show that var(Y) = E(var(Y|X)) + var(E(Y|X))
10. Let X, Y be iid uniform [0,1] rv's, and Z= X+Y. What is E(E(X|Z))? What is var(E(X|Z))?