Topics Outline for MA381 -

Text: Fundamentals of Probability with Stochastic Processes , by S. Ghahramani

**Ch. **** 1 Axioms of Probability **

Required:Ch. 1.1 - 1.4: Sample Space, Events, Axioms of Probability, Basic Theorems

Optional:Ch. 1.5 - 1.7

**Ch. **** 2 Combinatorial Methods **

Required:Basic Counting Principles, combinations, permutations (no more than 2 days)

**Ch. **** 3 Conditional Probability and
Independence **

Required:Ch. 3.1 - 3.5: Conditional Probability, laws of multiplication and total probability, Bayes' Formula, Independence

Optional: Ch. 3.6Applications of probability to genetics

**Ch. **** 4 Distribution Functions and Discrete
Random Variables **

Required:Ch. 4.1 - 4.6 Random Variables, distribution functions, discrete random variables, expectation of discrete random variables, variances and moments of discrete random variables, standardized random variables.

**Ch. **** 5 Special Discrete Distributions **

Required:Ch. 5.1 - first part of 5.3 Bernoulli and Binomial, Poisson (including Poisson Process), Geometric

Optional:Negative Binomial, hypergeometric

**Ch. **** 6 Continuous Random Variables **

Required:Ch. 6.1 - 6.3 Probability density functions, density function of a function of a random variable, expectation and variances

**Ch. **** 7 Special Continuous Distributions **

Required:Ch. 7.1 - 7.3 Uniform, Normal , Exponential

**Ch. **** 8 Bivariate Distributions **

Required:Ch. 8.1 - 8.3 Joint distributions, independent random variables, conditional distributions

Optional:Ch. 8.4 Transformations of 2 random variables

**Ch. **** 10 More expectations and variances **

Required:Ch. 10.1 - 10.4 expected values of sums of random variables, covariance, correlation, conditioning on random variables

**Ch. **** 11 Sums of independent random variables **

Required:Ch. 11.1 - 11.3, 11.5 Moment generating functions, sums of independent random variables, Markov and Chebyshev inequalities, Central Limit Theorem

**Supplement: Estimators and Confidence Intervals (do
as much as possible in no more than 3 days) **

- Estimators of mean µ , variance s 2 and Bernoulli parameter p
- Unbiased Estimators?
- Confidence intervals for mean µ given normal population with known variance
- Use of confidence intervals to do hypothesis testing
- Sample size calculations
- (optional) Confidence interval for difference of two means from different normal populations