Math with Statistics

Overlapping courses

A major in statistics requires the completion of calculus up to multivariable calculus, as well as Introduction to Linear Algebra (Math 250). Beyond this, there are a few other places where statistics and mathematics curricula overlap. Majors in both Statistics and Mathematics require an introductory computer science course, and several math courses can be used for statistics requirements.

Core statistics theory courses

The Department of Statistics offers courses for Theory of Probability (960:381) and Theory of Statistics (960:382), both of which are required for the major. The similarly-named courses in the Department of Mathematics, respectively Mathematical Theory of Probability (477) and Mathematical Theory of Statistics (481), can be used as a substitute for these courses. The mathematics courses count towards a major or minor in mathematics, but the statistics courses do not count.

Mathematics elective for Statistics

The Statistics major also requires an additional elective taken from the Mathematics department. This course must be either a differential equations course or a 300+ course (except 477 and 481).

Statistics/Mathematics Major

There is a Stat/Math major housed in the Department of Statistics in collaboration with the Department of Mathematics. This major requires Calculus I-IV (unlike the regular major, not just Calculus III) and Math 250, as well as the following 300+ math courses: 300, 311, 478. In addition, the Theory of Probability/Statistics requirement for the standard Statistics major functions the same way for the Stat/Math major, allowing for 477 and 481 to be counted towards the major.

Some advice for mathematical statistics

These are just the opinions of one contributor. There are plenty of viable options to go for, and the author disagreeing/disliking an option does not make it a wrong choice. Furthermore, this advice is meant for theory-minded students, rather than applied students.

Subjects to focus on

The most important subjects to focus on are:

  • Probability: You can just take 640:477. Taking 640:481 (Mathematical Theory of Statistics) is also fine, but because it does not have pre-requisite of real analysis, it is not easy to give a satisfactory introduction of the subject.
  • Real analysis: You can take 640:311, but I strongly recommend taking 640:411 as well. 640:412 is not as necessary but also highly useful. If you can, the author also strongly recommends 640:501 (first graduate course in real analysis).
  • Linear algebra: The author strongly recommends taking 640:350 (Linear Algebra). Taking 640:250 will usually not give enough experience with linear algebra; being familiar with this subject is extremely useful.

These classes are to me the bare minimum to have a complete understanding of the basic theory.

  • If you do not know measure theory: it is good to know this subject, but you can get around it without suffering from much. For example:
    • Probability densities are usually defined by the Radon–Nikodym theorem. However, you can still have a working understanding of them by simply assuming they exist. Though, you should keep a few examples in mind to remind yourself that not all probabilities have densities.
  • If you have not taken 640:350: Most of the linear algebra needed is for things like singular value decomposition, Moore–Penrose inverse, etc. but you can just learn these concepts as you go; if you have taken 640:250, the ideas should not be completely foreign.
  • If you have not taken 640:311/411: If a proof does not require regularity assumptions (i.e. continuity, differentiability, etc.), you can try to intuitively understand it by imposing these conditions and seeing if things make more sense.
  • If you have not taken 640:477: Take it!

The author has not taken many undergraduate statistics courses because their impression is that they are more applied. (For anyone else who knows more about them, it would be useful to contribute.)