# Harvard MS student probability vs real analysis for PhD

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Hello everyone,

I will be beginning biostat grad program at Harvard in the fall. I ultimately plan to apply to statistics/OR PhD programs next year (I will be completing my MS next fall).

I am aware of the fact that many graduate schools prefer students to have completed two semesters of real analysis, and I have been definitely planning to take graduate-level real analysis courses at Harvard graduate school. However, I come from a non-mathematics background (my degree is in chemical engineering from one of the ivies, and I have not taken proof-based math classes in undergrad), and my advisor suggests that I take proof-based probability classes that are designed for biostat PhD students in lieu of real analysis.

Descriptions of probability I and probability II are as shown below:

1) Probability I

Axiomatic foundations of probability, independence, conditional probability, joint distributions, transformations, moment generating functions, characteristic functions, moment inequalities, sampling distributions, modes of convergence and their interrelationships, laws of large numbers, central limit theorem, and stochastic processes.

2) Probability II

A foundational course in measure theoretic probability. Topics include measure theory, Lebesgue integration, product measure and Fubini's Theorem, Radon-Nikodym derivatives, conditional probability, conditional expectation, limit theorems on sequences of random stochastic processes, and weak convergence.

Would it be sufficient to take these probability classes in replacement of two semesters of real analysis? I plan to take one elective stochastic course as well in addition to two probability classes.

Thank you.

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I would take real analysis, since you don't have it on your transcript. Judging from these course descriptions, Probability I is just the first semester of the year-long sequence of Mathematical (Theoretical) Statistics I & II, so there does not seem to be any point in taking it separately from the typical Masters-level mathematical statistics sequence.

For Probability II at your institution, the instructor probably won't allow you to take it if you haven't taken a basic analysis course. And you can take this in your PhD program anyway (at most stat and biostat programs, this would be the required PhD-level course in Probability). Many PhD programs only require one semester of measure-theoretic probability theory (including reputable programs such as CMU and Duke), with an optional second semester of advanced probability theory for the students who plan to specialize more on probability theory than on statistics. I did my dissertation on theoretical stats, and I only needed to know the very basics of measure theory and probability theory for my research.

Edited by Applied Math to Stat
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The graduate level real analysis at Harvard math 212 a and b is extremely advanced and should probably only be taken by math grad students. Stat 210 and 212 should serve your needs. 210 is usually taught by Blitzstein who is an excellent teacher

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After looking at the course websites for Math 212a and Math 212b, I would agree with the above poster that these are not necessary for statistics Masters students.

It seems like Math 112 (Introductory Real Analysis) at Harvard would be the most appropriate real analysis class for Masters students who are interested in applying to stat/biostat PhD. It seems that only one semester of Math 112 is necessary too, since they seem to cover the whole analysis sequence in one semester (other schools go more slowly and spread it out over two semesters).

OP: are grad students allowed to take a class like Math112 at Harvard? If so, I would strongly recommend taking this.

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Grad students can take math 112.  Math 112 is the basic upper division real analysis class(Rudin or some variation of it). Math 114 is the standard graduate level analysis(Royden) at most schools.

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6 minutes ago, Gauss2017 said:

Grad students can take math 112.  Math 112 is the basic upper division real analysis class(Rudin or some variation of it). Math 114 is the standard graduate level analysis(Royden) at most schools.

In that case, the OP would be a very competitive candidate for PhD programs if s/he took both Math 112: Introductory Real Analysis and Math 114: Analysis II. I think just taking Math 112 might be sufficient for being in the discussion for admissions, but adcoms may look favorably upon having studied basic measure theory/Lebesgue integration. In my PhD program, we had to take basically a whole semester of measure theory/abstract integration, but a lot of stat programs condense this into the first few weeks of a one-semester Probability Theory class before diving into the classical probability theorems (Fubini, Fatou lemma, etc.).

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@Gauss2017 Thank you so much! Your insights have been very helpful. I did email Blitzstein a few days ago and got the permission to take Stat 210 in the fall. I've been told that he is an amazing professor that truly enables you to 'understand' probability, and your suggestion confirms once again that I should take his class during my first semester.

I pretty much have my first semester figured out at this moment with the inclusion of stat 210 and will talk to my advisor about the second/third semester once I start my program in the fall. Once again- thank you so much for your help.

@Applied Math to Stat Thank you so much! I concur that Math 112 might be sufficient for admissions. Math 112 is typically offered in spring, and math 114 is typically offered in the fall so I might take Math 114 during my last semester (fall 2019) after Math 112 if that is what my advisor suggests for admissions. I really appreciate your help.

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You may want to think about stat 212 for the spring instead of math 114.  Recently 212 has been taught by Strook from MIT and he does a good job. It is basically an advanced probability class with real analysis mixed in.  Other good classes are stat 149 and stat 186 if you can fit them in your schedule

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@Gauss2017 Thank you so much! I will keep that in my mind. Your suggested classes definitely seem to be good elective courses to take during my last semester.

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• 1 year later...

What do people think of Harvard's math 23-A for real analysis I going into a PhD program?

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48 minutes ago, kingduck said:

What do people think of Harvard's math 23-A for real analysis I going into a PhD program?

Literally any real analysis class will be fine.  Nobody needs two semesters or graduate level analysis. Learn some basics about convergence and properties of sets so you're not overwhelmed when you take a measure theoretic probability class and you're fine.

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Posted (edited)

If you haven't taken any proof-based courses, I don't think it's a good idea to take grad-level probability courses. I would start from basic real analysis courses. Ideally you need two courses in real analysis covering material such as metric space/norm, completeness of metric space, hölder's inequality/Minkowski inequality(this can be further extended to other spaces such as Lp space and probability space) , basic topology, continuity/uniform continuity in metric space and Riemann integration. It would be better if you had exposure to Lebesgue integration before taking grad-level probability courses since probability space is a special general measure space and understanding Lebesgue measure definitely helps.

Edited by Casorati

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