# CLT and Characteristic Functions

### Spring 2013

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

Instructor

Will Perkins

wperkins3@math.gatech.edu

Office: Skiles 017

Topics
• Sections: 7.1 - 7.5
• Notes: Characteristic and Generating Functions, Central Limit Theorem, Statistical Method
• Homework Problems: HW4: 1-7, 9
• Definitions and Theorems:
1. Convergence in Distribution
2. Characteristic Functions
3. Properties of Characteristic Functions
4. Characteristic Functions of Common Distributions
5. Central Limit Theorem
6. Statistical Method
7. Probability Generating Functions
8. Properties of Generating Functions
9. Convolutions of Sequences
10. Null hypothesis
11. Alternate Hypothesis
12. p-Value
13. Statistical Test
• Sample Questions:
1. Prove the Central Limit Theorem.
2. Which of the conditions of the CLT can be relaxed?
3. Say you flip a coin 400 times and it comes up heads 250 times. Would you say the coin is fair? Explain using the statistical method, and CLT.
4. Show that a properly centered and normalized Pois(x) RV converges in distribution to a Normal RV as x-> infty.
5. How would you calculate the number of samples needed in a political poll to have a margin of error of .03 with probability .95?
6. Compute the 2nd and 3rd moments of a Poisson RV
7. Say you take n independent samples from a Uniform [0,1] rv. In what range will the average fall with probability .95?
8. Show that the conditions of the Lindeberg-Feller CLT are satisfied by iid RV's with a finite mean and variance.
9. Construct a sequence of iid random variables that do not satisfy the CLT.
10. What are the advantages and disadvantages of characteristic functions vs. generating functions?