The Laplace transform and applications, Fourier series, boundary value problems for partial differential equations.

The Laplace transform and applications, Fourier series, boundary value problems for partial differential equations.

Finite dimensional vector spaces, inner product spaces, least squares, linear transformations, the spectral theorem for normal transformations. Applications to convex sets, positive matrices, difference equations.

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The Laplace transform and applications, Fourier series, boundary value problems for partial differential equations.

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This upper-division course provides a rigorous foundation in the core principles of both discrete and continuous probability theory. The curriculum begins by establishing the formal probability framework, encompassing conditional probability, Bayes' theorem, and the mechanics of independent events. Building upon these fundamentals, the coursework transitions into the study of random variables and joint distributions. Emphasis is placed on extracting meaningful characteristics from these models, focusing heavily on calculating and interpreting expectations, variance, and covariance.

The latter portion of the course shifts toward the critical study of asymptotic behavior and limit theorems. Students will examine the distinct notions of convergence in probability and convergence in distribution. These analytical concepts are then applied to establish and understand the field's most fundamental asymptotic results: the Law of Large Numbers and the Central Limit Theorem. Time permitting, various additional topics will be covered, e.g.: finer statistical properties such Large Deviations estimates; and an introduction to the theory of finite-state Markov chains. 

Linear algebra eigenvalues, eigenvectors, applications to linear systems, least squares, diagnolization, quadratic forms.