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.

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    – whuber
    Dec 12 '13 at 7:15

I'm sure this has been asked on this site before so I suggest you search a bit more thoroughly.

Here are some helpful questions that are related:

Probability theory books for self-study

Introduction to statistics for mathematicians

Standard reference for classical mathematical statistics?

Introduction to applied probability for pure mathematicians?

Some basic theory that will be quite accessible with your background would be "Statistical Inference" by Casella Berger. That is what we used in our first year of biostatistics graduate school.

PDF: http://math.nenu.edu.cn/uploads/soft/120716/Casella_Berger_Statistical_Inference.pdf

Amazon: http://www.amazon.com/Statistical-Inference-George-Casella/dp/0534243126

Once you feel comfortable with that material you might want to look into some more rigorous understanding of probability theory by looking into texts such as Durrett's Probability Theory and Examples. This is what we use in our advanced PhD level probability course:

PDF: http://www.math.duke.edu/~rtd/PTE/PTE4_1.pdf

Note to really understand Probability theory you will have to be fairly comfortable with real analysis. If you haven't had any exposure to real analysis I highly recommend Walter Rudin's Principle's of mathematical analysis:

Amazon: http://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X


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