# Generating and Characteristic Functions

### Fall 2012

#### M / W / F, 12:05 - 12:55 pm

Instructor

Will Perkins

wperkins3@math.gatech.edu

Office: Skiles 017

Office Hours: Wednesday 10 am - 12pm, or by appointment

Topics
• Sections: 5.1, 5.2, 5.3, 5.7, 5.8, 5.9
• Definitions:
1. Generating function of a sequence
2. Convolution of Sequences
3. Probability generating function
4. Differentiation of generating functions
5. Generating functions of constant, Bernoulli, Binomial, Poisson, Geometric
6. Moment generating function
7. Characteristic Function
8. Characteristic function of common distributions
• Theorems: 5.1.12, 5.1.13, 5.1.18, 5.1.23, 5.1.25, 5.3.1, 5.3.4, 5.7.3, 5.7.4, 5.7.5, 5.7.6, 5.7.8, 5.9.3, 5.9.5
• Sample Questions:
1. Prove the WLLN using characteristic functions.
2. Is the moment generating function of a random variable defined for all t?
3. How about the characteristic function?
4. What is the characteristic function of a binomial with parameters n , \lambda/n?
5. Show that if lambda is fixed Bin(n, lambda/n) converges in distribution to Poisson(lambda)
6. Show that Pois(n), properly normalized, converges in distribution to a Normal rv.
7. Find a closed form for the generating function of the sequence {1/n!}
8. What is the nth term of the convolution of the all 1's sequence with itself?
9. Let U be uniform [0,1] and X ~ Bin(n, U). What is the distribution of X?
10. Let Z be poisson(n) and X ~ Bin(Z, p). What is the distribution of X?
11. What is the second derivative of the generating function of a Poisson random variable evaluated at 0?
12. What is the generating function of the sequence 1, 1/2, 1/4, 1/8, ....
13. What is the probability generating function of a geometric random variable?