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advice for engineering student interested in probability

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Hi, thanks in advance. Over the past year I’ve become very interested in stats/applied math, especially probability/stochastic processes. I’ve been juggling the thought of applying to PhD programs for these areas. Could you kindly provide some advice/evaluation of my worthiness for top 10-20 programs? Here's my current profile.

DWM engineering PhD student at MIT/Stanford/Berkeley
have a national fellowship DOE/NDSEG/NSF
ugrad GPA: 3.9 (also in engineering)
grad GPA: 3.8
math GRE: would take after 6mo-1yr of practice
courses at MIT/Stanford/Berkeley:
ugrad level: linear algebra (A), probability (A), analysis (B+), complex analysis (B+), measure theory (A), functional analysis (A), PDE theory (A)
grad level (planned for next year): measure theory, functional analysis, PDE, year-long measure-theoretic probability sequence
The two B+ grades were from my very first theoretical math classes and I didn’t have the “math maturity” at the time. I spent considerable time self-studying after those two mediocre grades and managed to get good results in subsequent classes. I've completed an MS degree, but am still a few years from a PhD. I have two first-author journal publications in applied physics research.
I think I would try for stats PhD programs since I don’t have any abstract algebra background. Also, I’ll probably wait until I take the grad level math classes before applying. I know many people would be happy with my current school situation, but I just can’t go back to doing engineering after seeing what can be done with math.
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Stanford, MIT, and Berkeley are three of the top four schools for probability in the U.S. (the other being NYU). If you are already a student at one of these schools, it should be pretty easy to either transfer departments, or simply work with a professor whose work you are interested in. I do not recommend that you reapply, unless you have other reasons for changing institutions.

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I think you are fine - just remember to apply to a wide range of program and diversify the rankings a bit (so if you apply to let's say, 7 programs, I would not choose them among the top 7 programs but the top 15). Good luck!

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