CLT and Characteristic Functions
T / Th, 4:35 - 5:55 pm
Office: Skiles 017
- Sections: 7.1 - 7.5
- Notes: Characteristic and Generating Functions, Central Limit Theorem, Statistical Method
- Homework Problems: HW4: 1-7, 9
- Definitions and Theorems:
- Convergence in Distribution
- Characteristic Functions
- Properties of Characteristic Functions
- Characteristic Functions of Common Distributions
- Central Limit Theorem
- Statistical Method
- Probability Generating Functions
- Properties of Generating Functions
- Convolutions of Sequences
- Null hypothesis
- Alternate Hypothesis
- Statistical Test
- Sample Questions:
- Prove the Central Limit Theorem.
- Which of the conditions of the CLT can be relaxed?
- 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.
- Show that a properly centered and normalized Pois(x) RV converges in distribution to a Normal RV as x-> infty.
- How would you calculate the number of samples needed in a political poll to have a margin of error of .03 with probability .95?
- Compute the 2nd and 3rd moments of a Poisson RV
- Say you take n independent samples from a Uniform [0,1] rv. In what range will the average fall with probability .95?
- Show that the conditions of the Lindeberg-Feller CLT are satisfied by iid RV's with a finite mean and variance.
- Construct a sequence of iid random variables that do not satisfy the CLT.
- What are the advantages and disadvantages of characteristic functions vs. generating functions?