Math 0-1: Probability for Data Science & Machine Learning

Explore independence in probability by contrasting dependent variables like hair and gender with independent die and coin outcomes, and apply probability via p(a∩b)=p(a)p(b). Differentiate mutual independence from pairwise independence.

Introduce discrete random variables and probability distributions, define random variables, and survey Bernoulli, binomial, geometric, and Poisson distributions with real-world binary outcomes like coin tosses and ad clicks.

Explore discrete random variables and their distributions, including Bernoulli, binomial, geometric, Poisson, and categorical distributions, with PMFs, supports, and the role of theta as probability of success.

Learn how to derive CDFs from PDFs for continuous distributions, including uniform, exponential, and standard normal, using the fundamental theorem of calculus and practical numerical methods or tables.

Master change of variables for random variables, from discrete to continuous, by mapping y = h(x) and summing p(x) over h^{-1}(y) with die roll and Bernoulli examples.

Condition on random variables across discrete and continuous cases, derive conditional and joint distributions, marginalization, and Bayes' rule for probability and density functions.

Explore the multivariate normal distribution, also called mvn or vector gaussian, focusing on the vector x, the mean mu, the covariance sigma, and its compact pdf with a quadratic form.

Demonstrates that in a multivariate normal distribution, zero covariance between variables implies independence, and the joint pdf factorizes into independent marginals.

Derive that the sum z = x + y of jointly normal random variables is normal, with mean mu_x + mu_y and variance sigma_x^2 + sigma_y^2 + 2 rho sigma_x sigma_y.

Practice computing the mean and variance of the Bernoulli distribution from its pmf and expected value, noting the mean theta and variance theta(1−theta).

Examine the kurtosis of the normal distribution by standardizing x to z, computing the fourth moment, and using integration by parts to show kurtosis equals three.

Apply a concrete exercise that shows zero correlation does not imply independence by using x uniform on -1 to 1, y = x^2, and computing covariance.

Explore the distribution of a linear combination of jointly normal variables, derive its mean and variance, and preview moment generating functions for extending to more variables.

Explores generating functions, including the moment generating function and the characteristic function, and shows how they help compute sums of independent variables and advance toward the central limit theorem.

Illustrates an example where the moment generating function does not exist for a distribution with density 1/x^2 on x ≥ 1, showing the MGF diverges and distribution has no moments.

Explore sums of independent random variables by using the moment generating function, showing that the sum's mgf equals the product of individual mgfs, with Bernoulli and binomial examples.

Explore how to compute the distribution of the sum of independent random variables using mgfs and see how summing geometric variables yields the negative binomial distribution.

Explore moment generating functions and characteristic functions for random variables and vectors, including sums of independent or iid variables and the uniqueness theorem.

Explore monotonicity of expected value: if x ≤ y, then E[x] ≤ E[y]. Proof sets z = y − x and uses linearity and non-negativity for discrete and continuous cases.

Present Chebyshev inequality: P(|Y − μ| ≥ a) ≤ Var(Y)/a^2, and interpret how variance controls deviation from the mean, with a Markov-based path to the law of large numbers.

The weak law of large numbers states that for independent random variables with the same mean mu and finite variance, the sample mean converges in probability to mu as n grows.

Examine convergence with probability one, or almost surely convergence, where X_n converges almost surely to y and implies convergence in probability, with the strong law of large numbers as application.

Study the beta distribution, a 0-1 model for rates like click-through and conversion, with alpha and beta parameters and its relation to the beta and gamma functions in Bayesian analysis.