# Basic Probability

### Fall 2013

#### T / Th, 9:35 - 10:55 am

Instructor

Will Perkins

wperkins3@math.gatech.edu

Office: Skiles 017

Office Hours: Thursday 11 am - 1pm, or by appointment

Topics
• Sections: 1.1 - 1.5, 2.1, 2.3, 2.5, 3.1, 3.2, 3.5, 4.1
• Definitions:
1. Probability Space
2. Field, Sigma Field
3. Probability Measure
4. Conditional Probability
5. Independent Events
6. Random Variable
7. Distribution Function
8. Indicator Random Variable
9. Discrete Random Variable
10. Probability Mass Function
11. Continuous Random Variable
12. Probability Density Function
13. Random Vector
14. Joint Distribution Function
15. Independent Random Variables
16. Discrete Random Variables: Bernoulli, Poisson, Binomial, Geometric
17. Continuous Random Variables: Uniform, Normal, Exponential
• Theorems: 1.3.4, 1.3.5, 1.4.4, 2.1.6, 2.5.5, 4.2.3
• Sample Questions:
1. Are there random variables which are neither discrete nor continuous? If so, give an example.
2. Give an example of a field that is not a sigma field.
3. Show that the sum of two independent Poisson rv's is a Poisson.
4. Give an example of a collections of rv's that are pairwise independent but not jointly independent.
5. Calculate the probability that the sum of k independent exponential rv's with mean 1 is less than 1.
6. On the set of outcomes {1, 2, ... n} give three different sigma fields.
7. Say two probability spaces share the same outcome space and sigma field, but have different probability functions. Are the same events independent in both spaces? Why or why not?