# Help with understanding and studying probability

I'm a CS major and self studying sheldon ross's first course in probability book, before that I have taken a calculus based probability course, not a strong one, which ended with superficially covering content in ross's 6th and 7th chapter. We proved gamma(1/2) = pi, solved integrals for calculating moment generating functions of some common ditributions, joint pdfs, jacobian determinant etc.

The problem is, although doing a calculus based probability course and being comfortable with linear algebra and calculus I spend a lot of time understanding example problems in this book. My intention was studying some real analysis and measure theory and continue with stochastic calculus and I was hoping that at least at the end of summer I would get my feet wet in stochastic calculus. With this speed it seems that I would only be able to complete the ross's book at the end of summer if I attempt fair amount of exercises for each chapter.

As an example, I was reading sum of independent random variables and the author explains it by convolution and I lost lots of time (hours) just understanding the convolution integral and its applications to the problem but this is only a single example in the book and there are many examples of this kind.

I love to learn math and doing this as an hobby, but I have started to think that I am a bit stupid for learning math. What do you suggest me for studying math books efficiently ? And for professional math major students, do you spend a lot of time on text or do you grasp it very quickly ? Can you give some strategies for learning faster ?

As a note I'm a foreign student and do not have abundance of qualified instructors or tutors who would help me with points I do not understand.

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I believe I can help you at least with regards to probability. This field was very practical and applied but on the other hand quite informal. Kolmogorov made a strong formal theory from this field, however his approach was based on his level of math (or level of his colleagues).

While learning probability it's very useful to follow the red line - more precisely, to understand why do we need this definitions, why this theorem should be helpful, why this lemma should be right? That is the responsibility of the teacher/author of the book since they are experienced and students/readers are not. E.g. I am not sure at all that convolutions are so necessary at least to understand the notion of independence.

I strictly advise you the following book: Steve Roman, "Introduction to the mathematics of finance". Regardless of the title, there are 3 perfect chapters only on the probability theory.

Say, I had problems with understanding notions of measurability with respect to sigma-algebra or conditional expectation. Why? Because sigma-algebra was just given as a class of sets with certain properties and conditional expectation was introduced through the too formal definition. Meanwhile, in this book it was explicitly written what each operation/definition means.

Since you are writing here I would assume that you are not too lazy and you can learn from the books. It may be just a matter of the book itself. I haven't read the book by Ross, but usually "classical" books in probability miss that kind of explanation. They are good to read when you have a first impression about probability theory to make your knowledge strict and formal (IMHO).

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The only way by which you can, learn Math more faster and efficiently, is to devote more time to it.

• Note down the important results/theorems and try reproducing the proofs on your own.

• Ask yourself dumb questions. Prof. Terence Tao, has a written a very nice article.

• Solve all the exercises. By solving problems, one masters the subject and acquires problem solving skills.

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A great way to understand maths, and to me the most pleasant one, is to have someone to ask questions to and discuss your understanding of the material with, and ideally explain the stuff to you. If you have a math major friend who knows his way around probability, or know some professor at the university, you could ask them to help you a bit. For instance, in understanding the importance of the theorem, how people think about it and why it exists in the first place. You could get there completely on your own (?), but you would miss out on the wealth of knowledge people around you have already accumulated, waiting to be passed around in informal discussion.

So my advice would be to read up on your book and get to a point where you have learnt new material and have accumulated lots of questions, both mathematical and of other nature, and have someone shed light on your interrogations. Not everything will be aswered, but it should help you a great deal.

Have a great summer!

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And of course, posting such questions on this site would be helpful :) – Elliott Jun 10 '11 at 19:54

Two things:

(1) Given your background, I think there will be a very manageable number of "surprises" (like convolution integrals) in Ross that seriously sidetrack you. Just set a reasonable pace for yourself (how much material do you want to get through every week?) and have fun with it. It doesn't seem likely that Ross will take up your entire summer, especially since it sounds like you're pretty far along already.

(2) I usually read technical books by looking at an example, putting the book aside, and then trying to solve it myself. If I'm unable to solve the example, then I look at the next line of the explanation in the book (to get a hint) and then try again.

I find that it's very hard for me to understand (and even harder to retain) the explanation of an example if I haven't tried to solve it myself first. On the other hand, attempting the problem myself (and possibly even succeeding) gives the book's explanation a lot more context; anything from "oh, this is the key idea I was missing" to "interesting, the book solved this a completely different way".

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I think I can somehow give some recommendation on this. If you want to somehow get into stochastic calculus by the end of summer, I would recommend picking up a book on stochastic calculus that does not require extensive knowledge on measure theory(but terminologies like measurable, filtration will come in inevitably). When I took a course in stochastic calculus, the textbook was "Stochastic Calculus and Financial Application" by J. Michael Steele. I took this course when I had a bare minimum prereq in probability. The essential stuff that we have been using a lot of time were: Expectation, Conditional Expectation (super Important), Moments and Normal distribution. Of course there are a lot of technical details that you can pick up along the way.

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