What books would y'all recommend for someone who has taken a couple of college-level proof courses but never took anything on probability and statistics?
Best of luck. I can see from your post that you're thinking about performance tuning, I'm assuming you mean of software. That's a nice area - the nice part is that compared to fields like medical genetics, data on performance of software is relatively cheap to get, so a lot of issues about small sample sizes are surmountable.
I've said this elsewhere, but I recommend Casella & Berger's Statistical Inference. It will take you from probability theory to statistics... all the basics. I found it to be a very readable text, and it is used for many first graduate courses in stats - for me it was used for a 2 semester sequence first focusing on probability, then on statistics.
I like Casella and Berger, especially because it does so well at showing the connection between probability and statistics (first five chapters or so), but it should be noted that its approach is very different from the one in the slide deck above. Casella/Berger is purely frequentist rather than computational or Bayesian, and it spends a lot of time on likelihood theory and treats e.g. bootstrapping only very summarily.
I'd like to start with formalizing my knowledge of the basics -- hypothesis testing and confidence intervals. I do some performance tuning work, and it seems like a good idea to understand if my changes are making statistically significant improvements.
Those are what're thought of as standard "frequentist" basics; practical tools originally invented for practitioners. Many basic texts aiming to be statistics for scientists will do well, though unfortunately I don't have particular suggestions. Sorry about that!
