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I am deciding which of the following two subjects to take as one elective in the coming term.

  1. Complex Analysis
  2. Theory of Partial Differential Equations

Ideally I'd like to take both. However, I will have other things to do. Therefore, I want to take something that is direct relevant to statistics and probability. Any suggestion? Thank you!

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    $\begingroup$ not sure if this question is on topic but it is fun to answer so I would say 2. The fact is, it depends which area you will be working in, but I suspect you wouldn't know without a crystal ball. I think 2 is useful because sometimes the form of the function f that relates x and y are derived from PDEs. $\endgroup$
    – qoheleth
    Jul 10 '14 at 5:10
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    $\begingroup$ Is there a particular field in which you want to apply your statistics knowledge (other than "Statistics", of course)? Because you might want to consider that as a tiebreaker. Or, compare the professors teaching the courses--it won't matter how useful the course "should" be if you're stuck with a lousy lecturer (conversely, a great professor is worth their weight in gold!). $\endgroup$
    – Steve S
    Jul 10 '14 at 5:48
  • $\begingroup$ @SteveS I mostly deal with insurance data, such as, mortality and claim data. $\endgroup$
    – LaTeXFan
    Jul 10 '14 at 5:56
  • $\begingroup$ I don't know--if you're interested in the Insurance/Finance overlap then maybe go for PDE's since (Stochastic) PDE's are such a big part of Quantitative Finance... That's all I can come up with... Hopefully someone who knows what they're talking about will come along... $\endgroup$
    – Steve S
    Jul 10 '14 at 7:00
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If those are the choices, then Complex Analysis is probably the way to go. PDEs are interesting and useful, but not too much in statistics. Complex Analysis, on the other hand does pop up from time to time. Its most common application is in distribution theory, where the characteristic function is defined as: $$ \phi_X (t) = E\{e^{itX} \}.$$

There is an important theorem in distribution theory which states that there is a 1:1 correspondence between distribution functions and characteristic functions.

Now I personally don't think that by itself is a reason to do CA. But if it's that or PDE and you know you want to do mathematical statistics, then CA is the choice.

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  • $\begingroup$ And Characteristic functions are used to prove central limit theorem and for saddlepoint expansions $\endgroup$
    – seanv507
    Jul 10 '14 at 7:29
  • $\begingroup$ True, but you don't need hardly any complex analysis for those purposes. The inversion theorem is the big thing that needs CA. $\endgroup$
    – Dennis
    Jul 10 '14 at 15:33

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