Introduces the basic notions of Probability: random variables, expectation, conditioning, and the standard distributions (Binomial, Poisson, Exponential, Normal). This course also covers the Law of Large Numbers and Central Limit Theorem as they apply to statistical questions: sampling from a random distribution, estimation, and hypothesis testing.
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Course notes
Continuous Random Variables – MATH 3510
Definition of continuous random variables, probability density functions
Independent Random Variables – MATH 3510
Introduction and useful properties of independent random variables
Variance and Standard Deviation – MATH 3510
Definition of Variance, Standard deviation. Chebyshev’s Inequality
Can we trust the expected value? – MATH 3510
Exploring a concrete situation involving expected values and investments
Expected Value of Sums of Random Variables – MATH 3510
Computing expected value of sums of random variables and another formula compute expectations
Monty Hall Dilemma – MATH 3510
Using conditional probabilities to formalize the solution of the famous Monty Hall Dilemma
Conditional Probability – II – MATH 3510
Introduction to conditional probabilities – Part II
Continuity of Probability Measure – MATH 3510
Continuity of probability measure: increasing and decreasing sequences of events
Extending Axioms of Probability – MATH 3510
Extending the axioms of probability to sample spaces with infinitely many outcomes
Using Probability Distributions: exercises – MATH 3510
Some exercises whose solutions are made easier if you use some well-known probability distributions
Study Guide: Probability Distributions – MATH 3510
A study guide about the most important probability distributions and how to use them
Binomial Distribution: concrete problem – MATH 3510
Modeling concrete problems with binomial distributions
Binomial Distribution – MATH 3510
Introduction to the most important discrete distribution
Sample Space with equally likely outcomes – MATH 3510
Formulas and examples when we have a sample space with equally likely outcomes
Law of Large Numbers and Simulations – MATH 3510
Comments on Law of Large Numbers and simulations
Introduction to the course – MATH 3510
General comments about probability theory and our course!
