Provides preparation for graduate study in statistics by surveying topics in linear algebra, advanced calculus, mathematical analysis, and other areas chosen to strengthen students’ analytical and mathematical skills.
Athena Title
Tools for Statistical Theory
Prerequisite
Permission of department
Semester Course Offered
Offered fall
Grading System
A - F (Traditional)
Student learning Outcomes
Students will understand vector spaces, subspaces, linear transformations, and matrix operations, including inverses, rank, and eigenvalues, to support statistical theory.
Students will utilize matrix decompositions, quadratic forms, and singular-value decomposition in statistical modeling and estimation.
Students will demonstrate proficiency in limits, continuity, convex functions, and asymptotic notation, with applications to statistical functions.
Students will use differentiation, Taylor’s theorem, and optimization techniques to analyze functions relevant to statistical inference.
Students will establish convergence properties of sequences and series, including uniform convergence and power series, and apply them to statistical methods.
Students will utilize fundamental limit theorems such as the weak and strong laws of large numbers in statistical contexts.
Students will apply series expansions, probability generating functions, and Poisson approximations in statistical modeling and inference.
Students will synthesize concepts from linear algebra, analysis, and optimization to support rigorous statistical reasoning and theoretical developments.
Topical Outline
Vector spaces and subspaces
linear transformations
basic operations on matrices
the rank of a matrix
determinant
the inverse and the generalized inverse of a matrix
eigenvalues and eigenvectors
the diagonalization of a matrix
quadratic forms
the singular-value decomposition
inner product
orthogonal projection
orthonormal bases
the spectral theorem
real and complex numbers
supremum and infimum
limit of sequences
tests of convergence
metric space
open and closed sets
compact sets
completeness and the Cauchy criterion
continuity
differentiation
the Mean Value Theorem
Taylor’s theorem
multivariate differentiation
convex functions
maxima and minima of a function
uniform convergence of functions
power series
sequences and series of matrices
integration
selected applications to statistics
Institutional Competencies Learning Outcomes
Analytical Thinking
The ability to reason, interpret, analyze, and solve problems from a wide array of authentic contexts.
Critical Thinking
The ability to pursue and comprehensively evaluate information before accepting or establishing a conclusion, decision, or action.
