The preferred background for students entering the program is a bachelor's or a master's degree in engineering, science, or mathematics. Entering students will need to have studied the following topics in mathematics and statistics as preparation for core courses in the first year of the program. Therefore, applicants who have not yet studied these topics should register for the appropriate courses during the year in which they apply.

## Linear Algebra

At the level of the first two chapters in Strang, Linear Algebra and Its Applications, or the first three chapters in Strang, Introduction to Linear Algebra: matrix operations, linear transformations, rank, solving systems of linear equations. This material is often found in the linear algebra component of an undergraduate multivariate calculus course, so it may not be necessary to take a course devoted to linear algebra.

## Probability with Calculus

At the level of Ross, A First Course in Probability, Pitman, Probability, or Durrett, The Essentials of Probability: random variables, probability distributions including binomial, geometric, Poisson, exponential, and normal, independence, covariance, conditional probability, conditional expectation, the central limit theorem.

## Statistics with Calculus

At the level of Tamhane and Dunlop, Statistics and Data Analysis: From Elementary to Intermediate: sample mean and variance, confidence intervals and hypothesis testing for population means, variances and proportions, based on the normal distribution, the t distribution, the chi-square distribution, and the F distribution, and simple linear regression and correlation.

## Mathematical Proofs

It is best to take a theoretical mathematics course in which students learn to do rigorous proofs. If you do not have the opportunity to do so, we strongly advise you to carefully review Solow’s paperback text, How To Read and Do Proofs. A course in real analysis at the level of Rudin, Principles of Mathematical Analysis is the ideal preparation. However, it is not necessary to take it before enrolling in the program, because PhD students can take Math 321-1 Real Analysis at Northwestern in the fall of their first year.

## Computer Programming

Students should be familiar with computer programming in some language before enrolling in the program, but the particular language is not important. C++, Java, Python, and MATLAB are often used by our students.