Lec 1: Probability Models and Axioms
Lec 2: Conditioning and Bayes' Rule
Lec 3: Independence
Lec 4: Counting
Lec 5: Discrete Random Variables I
Lec 6: Discrete Random Variables II
Lec 7: Discrete Random Variables III
Lec 8: Continuous Random Variables
Lec 9: Multiple Continuous Random Variables
Lec 10: Continuous Bayes' Rule; Derived Distributions
Lec 11: Derived Distributions (ctd.); Covariance
Lec 12: Iterated Expectations
Lec 13: Bernoulli Process
Lec 14: Poisson Process I
Lec 15: Poisson Process II
Lec 16: Markov Chains I
Lec 17: Markov Chains II
Lec 18: Markov Chains III
Lec 19: Weak Law of Large Numbers
Lec 20: Central Limit Theorem
Lec 21: Bayesian Statistical Inference I
Lec 22: Bayesian Statistical Inference II
Lec 23: Classical Statistical Inference I
Lec 24; Classical Inference II
Lec 25: Classical Inference III
The proofs of these properties are both interesting and insightful. They illustrate the power of the third axiom, and its interaction with the remaining two axioms. When studying axiomatic probability theory, many deep consequences follow from merely these three axioms. In order to verify the monotonicity property, we set and
, where
for
. It is easy to see that the sets
are pairwise disjoint and
. Hence, we obtain from the third axiom that