Topics

- Sections: 1.4, 3.2, 3.6, 3.7, 4.2, 4.5, 4.6
- Definitions:
- Conditional Probability
- Independent Random Variables
- Joint Distribution Function
- Joint probability mass function
- Joint density function
- Joint Distribution
- Marginal distribution
- Conditional Distribution Function
- Conditonal PMF
- Conditonal Density Function
- 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:
- What is the conditional expectation of S_5 given that S_100 = 20?
- If X is Poisson with mean 1, what is E(X | X > 1)?
- What is E(S_n | S_n >0)?
- 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?
- Let X,Z be iid standard normals, and Y = X + Z. What is E(Y|X)? What is E(Y^2|Z)?
- If X and Y have Poisson distributions with mean 10, what is the distribution of X conditioned on X+Y?
- How would you define the conditional variance of Y given X? Show that var(Y) = E(var(Y|X)) + var(E(Y|X))
- 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))?