What should I study to best learn probability theory?

[markarroy]

I have the choice of taking, next spring, a course on linear algebra (theoretical, covers linear operator theory) or a course on applied math (covers asymptotic analysis and pdes). I cannot take both because i would be taking too many classes at once, and the classes i am already planning on taking will be the only time i can take them, ever. this will be the only time i can take applied math, but linear algebra can be taken in the fall of 2013.

On the applied side, I am strongly thinking about going into probability/related fields, which is built on measure theory and real analysis, and will probably take a course on measure theory in fall 2012 (again, the only time I can take it, or else I fall behind a lot in mathematics prereqs if I am going to take probability some time in my undergrad days.) I was wondering, if I am going to study advanced real analysis (dealing with baby rudin/kolmogorov) will I need a really strong knowledge of linear algebra beforehand? Will I need to understand many proofs in linear algebra to succeed in such a class?

Applied math seems to have a lot to do with any field i go into, and i don't want to miss taking that class. But is it possible to self study those topics if I have already mastered real analysis and linear algebra?

 

[micromass]

Don't worry about it, none of these things will probably be necessary to learn probability theory. At least, you won't encounter them soon.

I take it that you already know some linear algebra? Linear algebra is useful in multivariate things: multivariate distributions, differentials,... But a basic course in linear algebra should be enough to understand baby rudin and basic probability theory.

The applied math course you mention also won't be of much use for probability. I can see PDE's being used in stochastic equations, but you won't encounter those soon. Asymptotic analysis is important in numerical problems.

So, take what interests you the most. I don't think you will need either of those two courses soon.

Note: this reply is in the assumption that you already took a course of linear algebra. If you didn't, then you need to do so soon!

 

[markarroy]

Yes, I have already taken basic linear algebra, but no proofs at all.

I was wondering about a course in probability theory that follows after measure theory (similar to a grad course). I have also taken undergrad probability already. Does the same apply?

 

[micromass]

Yes, I have already taken basic linear algebra, but no proofs at all.

Hmm, no proofs, that changes things a bit. I'm still of the opinion that you won't need the extra course, but it wouldn't hurt taking it. Certainly because some proofs could be instructive.
Are you familiar with the following notions:
- linear map
- basis
- positive-definite
- diagonalization
- eigenvectors

If you are, then there's no need to take the advanced course. But could you list the topics in both the course you've already had and the course which you're going to take, to be sure?

I was wondering about a course in probability theory that follows after measure theory (similar to a grad course). I have also taken undergrad probability already. Does the same apply?

Yes, the same applies. (Pure) Probability theory uses a lot of notions from analysis and measure theory. You won't encounter linear algebra and applied math much.
Statistics is another things, this will probably use some notions of linear mathematics.
Stochastics will probably deal with some differential equations.
Can you also post the contents of this advanced probability course? Thanks!

 

[chiro]

I have the choice of taking, next spring, a course on linear algebra (theoretical, covers linear operator theory) or a course on applied math (covers asymptotic analysis and pdes). I cannot take both because i would be taking too many classes at once, and the classes i am already planning on taking will be the only time i can take them, ever. this will be the only time i can take applied math, but linear algebra can be taken in the fall of 2013.

On the applied side, I am strongly thinking about going into probability/related fields, which is built on measure theory and real analysis, and will probably take a course on measure theory in fall 2012 (again, the only time I can take it, or else I fall behind a lot in mathematics prereqs if I am going to take probability some time in my undergrad days.) I was wondering, if I am going to study advanced real analysis (dealing with baby rudin/kolmogorov) will I need a really strong knowledge of linear algebra beforehand? Will I need to understand many proofs in linear algebra to succeed irn such a class?

Applied math seems to have a lot to do with any field i go into, and i don't want to miss taking that class. But is it possible to self study those topics if I have already mastered real analysis and linear algebra?

I think if you have mastered real analysis and linear algebra, then a course in grad probability theory with all the sigma algebras and other stuff should be o.k. for you. You should have an "intuitive" understanding of probability, but since you said you took an undergrad course on it, then I don't see any reason why you couldn't do it.

I should note that linear algebra is really important for applied probability.

 

[markarroy]

thanks for the replies, i did a bit of looking and found i could take a course in the summer afterwards, so that means i don't have to wait (or cram) to take the courses

i think i get a good picture of it now

 

[creepypasta13]

I was wondering, if I am going to study advanced real analysis (dealing with baby rudin/kolmogorov) will I need a really strong knowledge of linear algebra beforehand? Will I need to understand many proofs in linear algebra to succeed in such a class?

Applied math seems to have a lot to do with any field i go into, and i don't want to miss taking that class. But is it possible to self study those topics if I have already mastered real analysis and linear algebra?

no, I don't think you need a *really* strong knowledge of linear algebra. I think a lower-div linear algebra course is ok. However, if this real analysis course is the first proof based class you've taken, then taking a proof-based linear algebra class first is more helpful, as you'll have prior 'mathematical maturity'