Probability and Statistics

Calculate probabilities of intersecting events, test for independence, and apply Bayesian updates.

Analyze discrete probability mass functions (PMFs), expected values, and variances.

Evaluate Probability Density Functions (PDFs), Cumulative Distribution Functions (CDFs), and calculate moments via integration.

Analyze multivariate random variables, marginalization, conditional distributions, and correlation.

Apply the Jacobian determinant for multivariate transformations and calculate the distribution of order statistics.

Master MGFs to find distribution moments and prove the Central Limit Theorem and Law of Large Numbers.

Derive statistical estimators using the Method of Moments and Maximum Likelihood Estimation (MLE).

Construct confidence intervals using Z/T statistics and evaluate hypotheses via p-values and significance levels.

Apply Bayesian inference to update probabilistic models using conjugate priors and posterior distributions.

Model relationships between variables using Ordinary Least Squares (OLS) regression.

Analyze systems that evolve probabilistically over time using transition matrices and steady-state distributions.