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I'd like to rebuild from the ground up and really make sure I have a strong grasp on probability and statistics.

I am able to compute answers in a pinch but I'd like a more formal understanding so I can really ensure I am being systematic about it. For example I have no idea what a probability mass/density function is.

Are there any good resources that go over all the basics and fundamentals while also providing answers to check my work against?

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A warm suggestion: Go online to half.com or amazon and look for a used statistics book. You don't need the latest version (expensive), you can also go for a previous version (=cheap, I would have given you mine for free if you had lived around the corner). My favorite: Introduction to statistics by Mario Triola, from which I taught the course couple of semesters to nursing students. I liked the book, good explanations and examples. (But there are other stats books as well). The answers (odds) are in the back, stats tables are with it too, and if not, those are available on the internet as well. You got everything in one book instead of bunch of PDF's. And if you get stuck on a problem, post it on MSE and we can help...Good luck!

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Are you interested in understanding probability theory/statistics or just using probability theory/statistics? You say "build from the ground up" and to me that means diving straight into measure theory. On the other hand, you might have never even taken a calculus class, which will make understanding what a probability density function is very difficult as, given a random variable $X:\Omega \to \mathbb{R}$ it is defined as the Radon-Nikodym derivative of a probability distribution (an induced measure) $\mathbb{P}(X^{-1}(\omega))$ with respect to $\mu$, the Lebesgue measure on $\mathcal{B}(\mathbb{R})$. For basic statistics, this is unnecessary but for a deeper understanding and for some applications, it is VERY necessary. Can you elaborate on your background/goals?

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  • $\begingroup$ Anything and everything. My calculus class in high school was not great (I can use derivatives and integration and set up problems, but I honestly have no idea how calculus works). I enjoy proving things that I otherwise take for granted, because I like feeling confident in knowing how and why something works. $\endgroup$
    – Sean Hill
    Commented Aug 4, 2016 at 17:48
  • $\begingroup$ Well in that case, your best bet is to get to the point where you do know how calculus works and then start out with some basic measure theory. You don't have to know everything in measure theory to understand probability theory or statistics but it gives you a coherent framework to understand how everything fits together. Basically, without measure theory, it is very awkward to define certain things like random variables or distributions. You can get a "fuzzy" understanding without it but it is hard to do any real work outside of the toy problems you find in intro level textbooks. $\endgroup$
    – Wavelet
    Commented Aug 4, 2016 at 18:00
  • $\begingroup$ What do you mean exactly? I always assumed a random variable was literally just a variable taken from some distribution where the likelihood of a particular value being drawn is equal to its corresponding value in the PDF? $\endgroup$
    – Sean Hill
    Commented Aug 4, 2016 at 18:02
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    $\begingroup$ "Well in that case, your best bet is to get to the point where you do know how calculus works and then start out with some basic measure theory." I disagree. It is clear from the OP's posts that he hasn't had much exposure to even calculus computations, and you're suggesting that the OP (essentially) learn real analysis, measure theory, and then do probability theory? Let the OP actually learn basic computations first before the OP decides to even touch real analysis. You don't need real analysis to do most applied probability problems. [continued] $\endgroup$ Commented Aug 4, 2016 at 18:08
  • $\begingroup$ [continued] I love probability and love the theory, and would never tell someone not even familiar with analysis to learn the measure-theoretic approach of probability the first time. $\endgroup$ Commented Aug 4, 2016 at 18:08

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