Would you please suggest a good, readable introductory probability book for graduate level ? I have Shiryaev 's Probability with me, however i want to find another one. Preferably with solution manual for self study.
I'm pleased to recommend Real Analysis and Probability by R. M. Dudley.
First of all, it seems to me that the author is very careful about the presentation of proofs. Every proof I encountered is as elegent as an elegent proof can be. Messy and intimidating proofs are deliberately avoided. One typical example is that the proof of Tychonoff's theorem in the book is the one using ultrafilter which consists of only two or three lines. Frankly speaking, I can't remain sane after reading those lengthy proofs, so I value this attribute of Dudley's textook very much.
Second, Dudley is very considerate to those readers (including me) who can't understand the subtlety of conditions of a theorem quickly. For instance, Dudley demonstrate that why the monotone condition is indispensable for Dini's theorem by an example, and my limited experience suggests that not every author would bother to do so.
Last but not least, the exercises are quite doable, even though you might not want to read the text in a linear order.
I $\heartsuit$ this book, so I can't resist the urge to recommend it to everyone.
Durrett "Probability: Theory and Examples" is usually the most recommended book. I am currently taking a graduate probability course and using this book. The solution manual is also available :)