Book recommendations for beginners about probability distributions I'm studying machine learning and every book I open I bump into chi-squared distribution, gamma-function, t-distribution, Gaussian, etc. 
Every book I have opened so far only defines what the distributions are: they don't explain or give the intuition on where the specific formulas for the functions come from. 
For example, why is chi-squared distribution the way it is? What is the t-distribution? What is the intuition behind the distribution? Proofs? etc. 
I would like to have a clear and fundamental understanding of the most commonly used distributions so that every time later on when I see them, I truly understand what is a t-distribution, what is a Gaussian distribution and most importantly why are they the way they are. 
It would be nice if the books / tutorials  can explain the concepts to a layman so that in order to understand them you don't already need to understand them x) Many books are like this, they don't fit for beginners :(
 A: You should read "Continuous univariate distributions" Vol. 1 & 2. by Johnson and Kotz.
Also "The Weibull distribution A Handbook" by Horst Rinne. Second one is a useful book to understand a distribution although this book focus on Weibull distribution. May be some material is not easy to under stand but early chapters give you some useful knowledge.  
A: For a short and easy overview of a lot of probability distributions, I recommend Probability and statistics EBook. Most distributions are described in chapter XV, but the more common ones are spread out in earlier parts of the book.
A: If you've no mathematical impediments there's a good overview in Ch. 3 of Casella & Berger, Statistical Inference, & much is covered in Grinstead & Snell, Introduction to Probability (it's free); for more detail I'd recommend Severini, Elements of Distribution Theory. But there are lots - it would be more difficult, I think, to find a less mathematical treatment that still gives the reader some feel for where different distributions come from.
