Probability Theory
A calculus-based first course in probability: sample spaces, events and Kolmogorov's axioms, counting methods for equally likely outcomes, conditional probability, independence and Bayes' theorem, discrete and continuous random variables with their pmfs, densities and cumulative distribution functions, expectation, variance and moments with linearity and indicator methods, the binomial, geometric, hypergeometric, Poisson, exponential, gamma, normal and beta families, joint distributions, covariance and correlation, functions of random variables and moment generating functions, the law of large numbers and the central limit theorem with the normal approximation, and a capstone on Markov chains and Monte Carlo simulation, with examples from insurance, polling, lotteries, transit and medical screening and runnable Python simulations throughout.
STAT-251 Year 2 3 credits 12 lessons · 4 h 49 min ▶ Video lessons coming soon
Offered by HSM University · Mathematics & Statistics
What you'll learn
- Describe random experiments with sample spaces and events, state Kolmogorov's axioms, and derive the complement, monotonicity, addition and inclusion–exclusion rules from them.
- Count outcomes with the multiplication rule, permutations, combinations and multinomial coefficients, and compute probabilities for equally likely outcomes, including lotteries, card hands and the birthday problem.
- Compute conditional probabilities, apply the multiplication rule and the law of total probability, use Bayes' theorem to revise probabilities in screening and classification problems, and test events for independence.
- Define discrete random variables through their pmfs and cdfs, and compute probabilities and distributions of simple functions of them.
- Work with continuous random variables through densities, cdfs, quantiles and the hazard function, and simulate from a distribution by the inverse transform method.
- Compute expectations, variances and moments, using linearity of expectation, indicator variables and the law of the unconscious statistician, and interpret skewness.
- Recognize and apply the Bernoulli, binomial, geometric, negative binomial, hypergeometric and Poisson distributions, including the Poisson approximation to the binomial.
- Recognize and apply the uniform, exponential, gamma and normal distributions, use the memoryless property and standardization, and read normal probabilities.
- Find marginal and conditional distributions from joint pmfs and densities, test independence, and compute covariances, correlations and variances of sums of dependent variables.
- Find the distributions of functions of random variables by the cdf and change-of-variable methods, compute convolutions, and use moment generating functions to find moments and identify sums.
- Prove the weak law of large numbers from Chebyshev's inequality, state and apply the central limit theorem, and use the normal approximation to the binomial with the continuity correction.
- Model processes with Markov chains, compute multi-step and stationary distributions, solve gambler's ruin by first-step analysis, and estimate quantities by Monte Carlo simulation with a standard error.
Course content
12 lessons · 4 h 49 min total · final exam
- 1 Sample spaces, events and the axioms of probability 25 min · quiz
- 2 Counting and equally likely outcomes 23 min · quiz
- 3 Conditional probability, independence and Bayes' theorem 22 min · quiz
- 4 Discrete random variables 24 min · quiz
- 5 Continuous random variables and densities 24 min · quiz
- 6 Expectation, variance and moments 24 min · quiz
- 7 Common discrete distributions 23 min · quiz
- 8 Common continuous distributions 25 min · quiz
- 9 Joint distributions, covariance and correlation 22 min · quiz
- 10 Functions of random variables and moment generating functions 27 min · quiz
- 11 The law of large numbers and the central limit theorem 28 min · quiz
- 12 Capstone: Markov chains and simulation 22 min · quiz
- E Final exam 20 questions
Requirements
Recommended background. You can start any course at any time.
Description
A calculus-based first course in probability: sample spaces, events and Kolmogorov's axioms, counting methods for equally likely outcomes, conditional probability, independence and Bayes' theorem, discrete and continuous random variables with their pmfs, densities and cumulative distribution functions, expectation, variance and moments with linearity and indicator methods, the binomial, geometric, hypergeometric, Poisson, exponential, gamma, normal and beta families, joint distributions, covariance and correlation, functions of random variables and moment generating functions, the law of large numbers and the central limit theorem with the normal approximation, and a capstone on Markov chains and Monte Carlo simulation, with examples from insurance, polling, lotteries, transit and medical screening and runnable Python simulations throughout.
Part of the Science stream, typically taken in year 2.