Learning Goals by Lecture

1 Lecture 1: Depicting Uncertainty

By the end of this lecture, you should be able to:

  • Identify probability as a proportion that converges to the truth as you collect more data.
  • Calculate probabilities using the Inclusion-Exclusion Principle, the Law of Total Probability, and probability distributions.
  • Convert between and interpret odds and probability.
  • Specify the usefulness of odds over probability.
  • Be aware that probability has multiple interpretations/philosophies.
  • Calculate and interpret mean, mode, entropy, variance, and standard deviation, mainly from a distribution.

2 Lecture 2: Parametric Families

  • Calculate expectations of a linear combination of random variables.
  • Match a physical process to a distribution family (Binomial, Geometric, Negative Binomial, Poisson, and Bernoulli).
  • Calculate probabilities, mean, and variance of a distribution belonging to a distribution family.
  • Find the probability mass function of a random variable transformation, e.g., \(X^2\).
  • Distinguish between a family of distributions and a distribution.
  • Identify whether a specification of parameters (such as mean and variance) is enough/too little/too much to specify a distribution from a family of distributions.

3 Lecture 3: Joint Probability

By the end of this lecture, you should be able to:

  • Calculate marginal distributions from a joint distribution of random variables.
  • Describe the probabilistic consequences of working with independent random variables.
  • Calculate and describe covariance in multivariate cases (i.e., with more than one random variable).
  • Calculate and describe two mainstream correlation metrics: Pearson’s correlation and Kendall’s \(\tau_K\).

4 Lecture 4: Conditional Probabilities

By the end of this lecture, you should be able to:

  • Calculate conditional distributions when given a full distribution.
  • Obtain the marginal mean from conditional means and marginal probabilities, using the Law of Total Expectation.
  • Use the Law of Total Probability to convert between conditional, marginal distributions, and joint distributions.
  • Compare and contrast independence versus conditional independence.

5 Lecture 5: Continuous Distributions

By the end of this lecture, you should be able to:

  • Differentiate between continuous and discrete random variables.
  • Interpret probability density functions and calculate probabilities from them.
  • Calculate and interpret probabilistic quantities (mean, quantiles, prediction intervals, etc.) for a continuous random variable.
  • Explain whether a function is a valid probability density function, cumulative distribution function, quantile function, and survival function.
  • Calculate quantiles from a cumulative distribution function, survival function, or quantile function.

6 Lecture 6: Common Distribution Families and Conditioning

By the end of this lecture, you should be able to:

  • Identify and apply common continuous distribution families.
  • Identify what makes a function a bivariate probability density function.
  • Compute probabilities from bivariate probability density functions.
  • Compute conditional distributions for continuous random variables.

7 Lecture 7: Maximum Likelihood Estimation

By the end of this lecture, you should be able to:

  • Explain the concept of a random sample.
  • Explain the concept of maximum likelihood estimation.
  • Define the likelihood function for a parametric model and recall why we often take its logarithm.
  • Apply maximum likelihood estimation for cases with one population parameter (i.e., univariate maximum likelihood estimation).
  • Use R to implement maximum likelihood estimation by an empirical approach.

8 Lecture 8: Simulation

By the end of this lecture, you should be able to:

  • Generate a random sample from a discrete distribution in both R and Python.
  • Reproduce the same random sample each time you re-run your code in either R or Python by setting the seed or random state.
  • Evaluate whether or not a set of observations are independent and identically distributed (iid).
  • Use simulation to approximate distribution properties (e.g., mean and variance) using empirical quantities, especially for random variables involving multiple other random variables.
  • Argue why simulations can approximate true properties of a stochastic quantity.