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I have Ross' A First Course on Probability and Bertseka's Introduction to Probability book. However these two books do not exactly give me what I am looking for.

The problem is Bertsekas' book is introductory, while Ross' book explains expression $E[\ E[X\mid Y] \ ]$ just by briefly saying it is a function of $y$ if I remember correctly.

I need a probability book which will make me feel comfortable with the symbols and letters rather than ready made formulas. I want to be able grasp when a random variable is a constant or when it is not. I want to be able to read and understand equations involving both random variables and constants and when are interpreted as a rv. when they are not etc.

cesim
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  • Try Probability with martingales by Williams – SBF Jan 11 '13 at 20:43
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    You used the letter (capital) $Y$ to refer presumably to a random variable, then you used the letter (lower-case) $y$, and it's not clear to me what you mean by it. – Michael Hardy Jan 11 '13 at 21:01
  • Try the course notes here which have replaced Ross's book as the text for an undergraduate course in probability. Incidentally, I doubt very much that Ross says anywhere that $E[E[X|Y]]$ is a function of $y$; he does point out that the expected value $E[X|Y = y]$ can be regarded as a random variable, denoted $E[X|Y]$, since it depends on our choice of the value $y$ that $Y$ took on; and so $E[X|Y]$ is a function of $Y$. – Dilip Sarwate Jan 11 '13 at 21:46

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If you want to truly understand the mathematical foundations try first learning a bit of measure theory. A probability space is a special kind of measure space and a random variable is a measurable function. Measure theory is not easy but there are several sources. Folland's book on analysis has a chapter on probability theory and Tao's recent book also related measure theory to probability.

Ittay Weiss
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Give Bruce Hajek's notes on Random Processes a try. I found some of the diagrams pretty intuitive and easy to understand even if you are a novice or not. See a snippet from his notes below: enter image description here

jay-sun
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