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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.
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merged by Tim Aug 30 '17 at 8:43

This question was merged with Book recommendations for probability because it is an exact duplicate of that question.

  • $\begingroup$ I agree that this is a duplicate to previous questions. I also first learned advanced probability out of Feller's two volume series. $\endgroup$ – Michael Chernick Mar 27 '17 at 12:23