Contenu de la matière

I. Probability Basics Reminder:
-    Introduction to probability spaces, events, and sample spaces.
-    Probability axioms and properties.
-    Combinatorics and counting techniques.
-    Conditional probability and Bayes' theorem.
-    Independence of events.
II. Random Variables:
-    Discrete and continuous random variables.
-    Probability mass functions (PMFs) and probability density functions (PDFs).
-    Cumulative distribution functions (CDFs).
-    Expected value, variance, and moments.
-    Joint probability distributions and marginalization.
III. Conditional Probability and Independence:
-    Conditional probability and conditional distributions.
-    Conditional expectation.
-    Conditional independence and its properties.
-    Markov chains and hidden Markov models.
IV. Probability Distributions:
-    Bernoulli, binomial, and multinomial distributions.
-    Poisson distribution.
-    Gaussian (normal) distribution.
-    Exponential and gamma distributions.
-    Joint and marginal distributions.
V. Central Limit Theorem:
-    Statement and applications of the central limit theorem.
-    Sampling distributions and the law of large numbers.
-    Confidence intervals and hypothesis testing.
VI. Statistical Inference:
-    Estimation theory: point estimation, maximum likelihood estimation.
-    Confidence intervals and hypothesis testing.
-    Parametric and non-parametric tests.
-    Bayesian inference.
VII. Random Processes:
-    Introduction to stochastic processes.
-    Discrete and continuous-time Markov chains.
-    Poisson processes and renewal theory.
-    Brownian motion and random walks.