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In the next months I am planning to deliver a (more-or-less) advanced course in probability theory. My students will have had already a first encounter with discrete probability theory (discrete random variables, Markov, Chebyschev, etc.) and a full course on Measure Theory.

In the past I have deliver the course making a lot of emphasis in the difference between discrete and continuous (in the second case, proving Radon-Nidokym theorem), but there is always the complaint of having a unified setting, which would require of the theory of tempered distributions.

My question: is there some good reference on probability theory in which the unification of discrete and continuous setting is given? I want to explain maybe in a couple of hours a gentle introduction to all of that, but I do not want to make my course a course on analysis, as I am very concerned on giving students a real probabilistic intuition of the subject.

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    $\begingroup$ How does the unification of discrete and continuous settings need tempered distributions? Isn't that already covered by basic measure-theoretic probability (and therefore almost every modern probability book)? $\endgroup$
    – unwissen
    Commented Jul 14 at 14:44
  • $\begingroup$ my impression, maybe i am wrong is that in several probability courses 1) one starts with discrete distributions (not talking about Dirac's delta distribution) and 2) after that you move to absolute continous distributions...but what is done combining both worlds is not treated. For instance in Grimmet Stirzaker my concern is not treated $\endgroup$
    – Johnny Cage
    Commented Jul 14 at 14:50
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    $\begingroup$ I think you are just partially wrong. Dirac's delta distribution $\delta_x$ is a discrete distribution, namely the distribution of a random variable $X$ which is equal to $x$ almost surely. I don't know Grimmet-Stirzaker, but if you mean "Probability and Random Processes" by Grimmet-Stirzaker, then this book of courses discusses general (real-valued) random variables and their distributions. But after the introduction of (general) probability measures they actually make a distinction between discrete and continuous (or anything which is not one of both) distributions that is not necessary. $\endgroup$
    – unwissen
    Commented Jul 14 at 15:24
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    $\begingroup$ I personally like "Probability and Stochastics" by Erhan Çinlar as a broad introduction to measure-theoretic probability (especially if the students already have a measure-theory background). There are no articifical distinctions (to not introduce measure-theoretic concepts) and a lot of heuristics and useful theorems are given. But in the end there are a lot of measure-theoretic probability books and you have to decide on what you want to focus content-wise in your course. $\endgroup$
    – unwissen
    Commented Jul 14 at 15:26
  • $\begingroup$ It seems that two conventions are out there: (1) A continuous distribution is one whose c.d.f. is continuous, and (2) A continuous distribution is one that has a density. But the Cantor distribution has a continuous c.d.f. and neither has a density nor is a mixture of any that has a density with anything else. I prefer the first definition. $\endgroup$
    – Michael Hardy
    Commented Jul 15 at 18:54

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