# Expectation and Variance

### Spring 2013

#### T / Th, 4:35 - 5:55 pm

Instructor

Will Perkins

wperkins3@math.gatech.edu

Office: Skiles 017

Topics
• Sections: 3.3, 3.4, 4.3, 5.6
• Notes: Expectation, Variance, First-moment method, Second-moment method
• Homework Problems: HW2: 1-6, 8-9.
• Definitions and Theorems:
1. Expectation
2. Properties of Expectation
3. Variance
4. Higher Moments
5. Markov's Inequality
6. Chebyshev's Inequality
7. First-moment method
8. Second-moment method
9. Covariance
10. Correlation
11. Indicator Random Variables
12. Expectation and Variance of Counting Random Variables
• Sample Questions:
1. Let p be a uniform [0,1] rv, then let X ~ Bin[n,p]. What is cov(X,p)?
2. Let Omega be a disc around the orgin in the plane of area 1, and let P be the Lebesgue measure. Let X be the x-coordinate of an outcome, and R the distance from the origin. Are X and R independent? What is their covariance?
3. Find a sharp threshold for the longest sequence of consecutive heads in n flips of a fair coin.
4. Give an analogue of Chebyshev's Inequality if we know the expectation of Exp[ \lambda X].
5. Find the asymptotics of the maximum of n independent normal RV's.
6. Prove that whp a simple symmetric random walk will not cross 0 between steps n and n+n^{1/3}.
7. Prove the Weak Law of Large Numbers.
8. Should positive correlation between random variables lead to greater or lesser concentration of their sum? Why?
9. Throw m balls into n bins at random. What is the covariance of the number of balls that land in bin 1 and bin 2? What is the expected number of empty bins? What is the variance of the number of emprty bins?