## Single- and multi-variable calculus

Approximately 4 quarters required

### Single variable

- Derivatives of polynomial, rational, trigonometric, exponential, and logarithmic functions
- Maximum-minimum problems
- Curve sketching, and other applications (velocity, acceleration)
- Antiderivatives and simple motion problems.
- Definite integrals,
- Integral tables and basic techniques of integration
- Polar coordinates
- Applications to areas, volumes, force, work
- Growth and decay problems.
- Indeterminate forms
- Improper integrals
- Sequences and series, especially Taylor's formula and power series

### Multi variable

- Vector functions and curves in two and three dimensions
- Partial derivatives
- Gradients
- Directional derivatives
- Multiple integrals in rectangular, polar, cylindrical, and spherical coordinates
- Physical and geometric applications

## Linear algebra

One course required

- Mathematical operations with matrices (addition, multiplication)
- Matrix inverses and determinants
- Solving systems of equations with matrices
- Euclidean vector spaces
- Eigenvalues and eigenvectors
- Orthogonal matrices
- Positive definite matrices
- Linear transformations
- Projections
- Linear dependence and independence
- Singular value decomposition

## Mathematical statistics

One course required; two quarters or one year strongly preferred

- Combinatorics and basic set theory notation
- Probability definitions and properties
- Common discrete and continuous distributions
- Bivariate distributions
- Conditional probability
- Random variables, expectation, variance
- Univariate and bivariate transformations
- Convergence of random variables: in probability, in distribution, almost sure
- Central Limit Theorem, Laws of Large Numbers
- Estimation: bias, MSE, consistency, sufficiency, maximum likelihood, method of moments, UMVUE, Rao-Blackwell Theorem, Fisher Information
- Hypothesis testing: significance level and power, Neyman-Pearson lemma, Likelihood ratio tests
- Confidence Intervals: definitions, duality with hypothesis tests

## Applied statistics

At least one applied statistics course is preferred, but not required.

## Advanced calculus

M.S. applicants interested in the Ph.D. are strongly encouraged to take advanced calculus (sometimes called "mathematical analysis" or "real analysis");

Applicants to the Ph.D. program **must** have this prerequisite.

- Sequences and series
- Continuity
- Limits of functions
- Uniform continuity
- Differentiation
- Taylor's theorem
- Lim sups and lim infs
- Riemann integration
- Basic methods of mathematical proofs, including epsilon-delta proofs, proof by contradiction, and mathematical induction.