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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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    $\begingroup$ While this is the go-to reference book: Continuous Univariate Distributions by Johnson, Kotz, for your purpose, any general undergrad stats books like Casella, Berger; Rohatgi will be suffice for they provide sufficient treatment in explaining the common distributions that are frequent in the studies. $\endgroup$ Dec 19, 2022 at 16:46
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    $\begingroup$ Generally one studies probability theory before studying statistics/machine learning. $\endgroup$
    – utobi
    Dec 19, 2022 at 17:26
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    $\begingroup$ @utobi That doesn't mean it's the best approach. A couple decades ago I taught an intermediate level course on prob & stats that developed the probability only when it was needed to address statistical problems. Based on pre- and post-testing results, this course was much more successful for learning statistics than the traditional method. $\endgroup$
    – whuber
    Dec 19, 2022 at 18:20
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    $\begingroup$ @whuber incredible! I am too curious, I wish I can have a look at these notes... $\endgroup$
    – utobi
    Dec 20, 2022 at 15:31
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    $\begingroup$ @utobi I was going to say there aren't many notes, but I checked and I would be wrong. I have records of my plans for each week of class; lab notes; many references to books and papers; and 140 pages of responses to student questions ("FAQs"). Students were required to write at least one thoughtful question on an "exit card" before leaving each class meeting. I collected those questions and provided extensive answers in the next meeting. These were good students, so the FAQs cover a lot of material -- far more than the few chapters of the textbook we made it through. $\endgroup$
    – whuber
    Dec 20, 2022 at 17:26

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I am not familiar with any book that would meet your requirements, but there's a Harvard course Statistics 110 by Joe Blitzstein that focuses exactly on the intuitions behind the distributions and probability theory concepts. (It is freely available online.)

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Blitzstein & Hwang's "Introduction to Probability" is in the same vein as the online course referenced by Tim but a lot more breadth. Highly recommended.

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