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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 :(

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    $\begingroup$ Most undergraduate texts on theoretical statistics or probability theory have a chapter on distribution theory that covers these questions. But how much mathematical background would you want to assume? $\endgroup$ – Scortchi Jul 2 '13 at 10:05
  • $\begingroup$ undergraduate mathematical background :) The fundamental building blocks. Is that sufficient? What kind of level of mathematics should I acquire before learning about the distributions? I have read a basic book about statistics, which only shortly presented the distributions I described in the question. $\endgroup$ – jjepsuomi Jul 2 '13 at 10:07
  • $\begingroup$ Some probability theory & calculus ought to do it - it depends how deep you want to go. $\endgroup$ – Scortchi Jul 2 '13 at 11:35
  • $\begingroup$ Okay, thank you :) Mostly I'd just want to understand what I'm doing $\endgroup$ – jjepsuomi Jul 2 '13 at 11:48
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    $\begingroup$ You also might find useful the references posted in this thread: stats.stackexchange.com/questions/56385/…. $\endgroup$ – Andre Silva Jul 2 '13 at 13:02
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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.

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  • $\begingroup$ why it doesn't have "probability mass function" ? $\endgroup$ – Woeitg Oct 3 '16 at 21:36
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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.

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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.

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