What is a good book to gradually improve problem solving and understanding in real analysis and probabilities?

Question

To give you a little context, I study for a master's degree on statistics and probability. I was never top in my class and I managed to get decent grades with as little work as possible. While writing my bachelor's thesis, I worked a little bit more on understanding a specific problem my professor gave me, and proved a small result regarding ramification chains using measure theory. It made a positive impression and got me fairly easy admitted to the master's programme. Now during my exams I realise I lack knowledge and experience regarding problem solving due to my laziness. I developed real interest in mathematics and I wish to improve at problem solving ( both theoretical and practical ).

I am looking for books concerned with problems and applications and that also provide explanation and/or solutions to them. It would be great to also point out theory that is needed not necessarily explaining it. The topics I am interested in are real analysis, probability and statistics. I believe those are fairly connected and I am looking for a mix of problems to gradually improve my understanding. It should start fairly easy and get to decent or even hard problems. It should cover most common concepts.

I thank you for your time!

P.S: I had an exam in an introductory course on statistics and almost failed miserably because even though I knew what expectation of a random variable was, I could not compute it since I did not have any relevant experience with series.

Answer 1

Suggestion: A Problem Book in Real Analysis.

Analysis is a profound subject; it is neither easy to understand nor summarize. However, Real Analysis can be discovered by solving problems. This book aims to give independent students the opportunity to discover Real Analysis by themselves through problem solving. (From preface)

Each chapter is headed by a portrait of some mathematician and an appropriate quotation. Next comes a brief list of basic definitions and theorems. The exercises themselves include both computational and conceptual items and vary in difficulty. (From MAA)

Answer 2

If you already know the basics concept about integration and differentiation you can first go with Calculus by Thomas & Finney or with Elements of Real Analysis by Narayan & Raisinghania. I would prefer this last one because it has the ability to explain concepts in a simple way but also rich in details.

When you finished it, or if you are already familiar with the topics in there, you can go with Real Analysis by Howie.

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About the probability, I'm studying it on Handbook of Probability by Florescu & Tudor and on A First Course in Probability by Ross. I like them because they are rich of examples, exercises and solutions, which are indeed important for the understanding of probability.

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Links (in order of appearing):

https://www.pearson.com/us/higher-education/product/Thomas-Calculus-9th-Edition/9780201531749.html

https://www.amazon.in/Elements-Real-Analysis-Narayan-Shanti/dp/8121903068

https://www.amazon.com/Real-Analysis-John-M-Howie/dp/1852333146/ref=sr_1_3?ie=UTF8&qid=1549116994&sr=8-3&keywords=Real+Analysis+howie

https://www.amazon.it/Handbook-Probability-Ionut-Florescu/dp/0470647272

http://julio.staff.ipb.ac.id/files/2015/02/Ross_8th_ed_English.pdf