Reddit reviews Probability and Random Processes

CLRS for algorithms/CS.

Probability and random processes for statistics.

Biological Sequence Analysis by Richard Durbin for my subfield of bioinformatics.

Uhm. Do you mean this one?

https://www.amazon.com/Probability-Random-Processes-Geoffrey-Grimmett/dp/0198572220

Probability and Random Processes is a wonderful book in probability; but focused on probability to the point that major statistical distributions (chi-squared, T) are merely asides.

Grimmett and Stirzaker.

The go-to book for the people I know is always http://www.amazon.com/Probability-Random-Processes-Geoffrey-Grimmett/dp/0198572220 , though this is perhaps a bit more stochastic-process-ish than what you are after (it is, however, exceptionally well-written).

If you're looking for a thorough and rigorous introduction into probability theory, I'd recommend going with Introduction to Probability Theory and Its Applications Vol.1 and 2 by Feller. Another well recommended book is Probability and Random Processes by Grimmett and Stirzaker (this starts from the get-go with measure theory).

If you're looking for general statistics, then you may want to look at All of Statistics by Wasserman and perhaps Bayesian Data Analysis by Gelman, et al.

Finally, since you're a physicist, you'll probably want to take a look at Monte Carlo methods in particular, such as with Monte Carlo Statistical Methods by Robert and Casella.

An oldie but good is Introductory Probability and Statistical Applications by Meyer! I've used newer, more fashionable textbooks (Ross, Miller & Miller) but this one is my favorite for the introductory level. It feels a bit dated at times (e.g. "although we cannot expect all readers to own a personal computer"), but the relevant math hasn't changed much in the passed few decades. It is very clear with more exposition than I've found in the newer books I mentioned.

As for an advanced text, I've heard good things about Probability and Random Processes by Grimmet and Stirzaker. My friends used it in a graduate course on basic probability, and compare it favorably to their undergraduate experiences with probability.

Probability and Random Processes by Grimmett is a good introduction to probability.

Mathematical Statistics by Wackerly is a comprehensive introduction to basic statistics.

Probability and Statistical Inference by Nitis goes into the statistical theory from heavier probability background.

The first two are fairly basic and the last is more involved but probably contains very few applied techniques.