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I have taken one introductory statistics class and I am finishing up an econometrics class, but these classes are more about applications and just applying formulas without any explanations as to how these formulas came about. I am looking for a good and thorough textbook or book on statistical and probability theory where the author explains the beginnings, the theory, the intuition behind these formulas. Not just proofs but exactly how were these formulas born.
i.e: How did the formula to calculate variance come about? How was this formula derived? What exactly and intuitively are degrees of freedom and how did the concept come about? Where did probability distributions arise? How were all these statistical formulas thought up in the first place? These are just a few of many questions.
My mathematics background includes: multivariable calc, linear algebra, intro stats, discrete mathematics.