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:
- Random Variable
- Distribution Function
- Change of Variables
- Discrete Random Variables
- Probability Mass Function
- Continuous Random Variables
- Density Function
- Independent Random Variables
- Common Distributions: Bernoulli, Binomial, Geometric, Poisson, Exponential, Uniform, Normal
- Joint Distributions
- Marginal Distributions
- Conditional Distributions

- Sample Questions:
- Calculate the 4th moment of a standard Normal rv.
- What is the distribution of the sum of two iid geometric rv's?
- Give an example of RV's that are uncorrelated but not independent.
- Prove that a distribution function can have at most a countable number of discontinuities.
- Find the distribution of the maximum of n independent Uniform [0,1] RV's.
- 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?
- 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.
- Find the asymptotics of Pr[Bin(n,p)= cn] for c= p and c != p
- 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?