I have done my masters in Statistics and am well acquainted with probability, though I am out of touch for last few years. Currently I am preparing for an exam which has a paper for probability and the syllabus is given below. I would prefer to follow a single book and make my own notes from it. I don't need over simplified starters guide kind of thing. I would prefer some of the classical books on the subject. I used to follow William Feller in my college days but am not sure if it covers everything in this syllabus (I don't have a copy with me right now). Thanks in advance for all your suggestions.
Syllabus
- Sample space and events
- probability measure and probability space
- random variable as a measurable function
- distribution function of a random variable
- discrete and continuous-type random variable
- probability mass function
- probability density function
- vector-valued random variable
- marginal and conditional distributions
- stochastic independence of events and of random variables
- expectation and moments of a random variable
- conditional expectation
- convergence of a sequence of random variable in distribution, in probability, in p-th mean and almost everywhere, their criteria and inter-relations
- Chebyshev's inequality and Khintchine's weak law of large numbers
- strong law of large numbers and Kolmogoroff's theorems
- probability generating function, moment generating function, characteristic function, inversion theorem
- Linderberg and Levy forms of central limit theorem
- standard discrete and continuous probability distributions.
