# Random Variables

### Spring 2013

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

Instructor

Will Perkins

wperkins3@math.gatech.edu

Office: Skiles 017

Topics
• Sections: 2.1, 2.3 -2.5, 3.1, 3.2, 3.5 - 3.7, 4.1, 4.2, 4.4 - 4.7
• Notes: Random Variables
• Homework Problems: HW 1: 3, 5. HW 2: 7
• Definitions and Theorems:
1. Random Variable
2. Distribution Function
3. Change of Variables
4. Discrete Random Variables
5. Probability Mass Function
6. Continuous Random Variables
7. Density Function
8. Independent Random Variables
9. Common Distributions: Bernoulli, Binomial, Geometric, Poisson, Exponential, Uniform, Normal
10. Joint Distributions
11. Marginal Distributions
12. Conditional Distributions
• Sample Questions:
1. Calculate the 4th moment of a standard Normal rv.
2. What is the distribution of the sum of two iid geometric rv's?
3. Give an example of RV's that are uncorrelated but not independent.
4. Prove that a distribution function can have at most a countable number of discontinuities.
5. Find the distribution of the maximum of n independent Uniform [0,1] RV's.
6. Let X_1, X_2, ... be independent Exponential RV's with mean \mu. Let S_k = X_1 + X_2 + ... + X_k. What is the probability that S_k < 1 but S_{k+1} > 1?
7. Let X_1, X_2 be iid standard normals. Let Y_1 = X_1 + 2 X_2, Y_2 = X_1 + X_2. Fin the joint density function of Y_1, Y_2.
8. Find the asymptotics of Pr[Bin(n,p)= cn] for c= p and c != p
9. Let X be the number of heads in n flips of a coin, and let Y be the indicator of a head on the first flip. What is the conditional distribution of Y given X? of X given Y?