Topics

- Sections: 5.1, 5.2, 5.3, 5.7, 5.8, 5.9
- Definitions:
- Generating function of a sequence
- Convolution of Sequences
- Probability generating function
- Differentiation of generating functions
- Generating functions of constant, Bernoulli, Binomial, Poisson, Geometric
- Moment generating function
- Characteristic Function
- Characteristic function of common distributions
- Central Limit Theorem

- 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:
- State the CLT. Which of the conditions are necessary? Can you weaken any of the conditions?
- Prove the WLLN using characteristic functions.
- Is the moment generating function of a random variable defined for all t?
- How about the characteristic function?
- What is the characteristic function of a binomial with parameters n , \lambda/n?
- Show that if lambda is fixed Bin(n, lambda/n) converges in distribution to Poisson(lambda)
- Show that Pois(n), properly normalized, converges in distribution to a Normal rv.
- Find a closed form for the generating function of the sequence {1/n!}
- What is the nth term of the convolution of the all 1's sequence with itself?
- Let U be uniform [0,1] and X ~ Bin(n, U). What is the distribution of X?
- Let Z be poisson(n) and X ~ Bin(Z, p). What is the distribution of X?
- What is the second derivative of the generating function of a Poisson random variable evaluated at 0?
- What is the generating function of the sequence 1, 1/2, 1/4, 1/8, ....
- What is the probability generating function of a geometric random variable?