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I have looked really hard, in vain, for a statistics textbook that condescends to explain (to the non-expert) how the $t$-distribution was arrived at.

By this I mean a well-motivated mathematical derivation that connects the $t$-distribution to the original problem that Gosset, Fisher, and others were grappling with when they settled on it.

Statistics textbooks treat this subject as if it were pure kryptonite. Just shocking.

If someone knows of such a book, please let me know.

NOTE: this question is a reference request. I'm not looking for such well-motivated mathematical derivation here. Rather, I'm looking for a pointer to such a derivation in a book or article.

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  • $\begingroup$ e.g. math.stackexchange.com/questions/474733/… $\endgroup$
    – Tim
    Commented Aug 28, 2017 at 9:56
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    $\begingroup$ Fisher's geometric arguments are worth a look. e.g. Here's Fisher, R. A. (1925). "Applications of "Student’s" distribution" (PDF). Metron. 5: 90–104. $\endgroup$
    – Glen_b
    Commented Aug 28, 2017 at 10:08
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    $\begingroup$ Why not read Gosset's original paper? It is quite readable: it uses fairly modern language, is not mathematically abstruse or sophisticated, is expansive, and even includes a Monte Carlo simulation! It is reproduced in Kotz & Johnson, Breakthroughs in Statistics Volume II: Methodology and Distribution, Springer-Verlag (1992). A short prefatory note by E. L. Lehmann puts the paper in context and references follow-up work by Fisher that set these results on a rigorous mathematical footing. $\endgroup$
    – whuber ♦
    Commented Aug 28, 2017 at 13:43
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    $\begingroup$ Mathematical derivations do appear here and are likely to be relevant for many readers. Here's one thread about them: stats.stackexchange.com/questions/151854/… $\endgroup$
    – whuber ♦
    Commented Aug 30, 2017 at 17:50

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This is developed in many books on mathematical statistics. It is in Bickel & Doksum https://www.amazon.com/Mathematical-Statistics-Basic-Selected-Topics/dp/0132306379 and it must be in Casella & Berger https://www.amazon.com/Statistical-Inference-George-Casella/dp/0534243126/ref=sr_1_1?s=books&ie=UTF8&qid=1503914420&sr=1-1&keywords=casella+berger But maybe you are looking for something more elementary ...

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  • $\begingroup$ FWIW: I just got Casella & Berger from the library, and I can't find anything even remotely close to what I'm looking for in it. In fact, I can't even find "$t$-distribution" (or anything like it) in the index! In light of this, I don't have much hope for Casella & Berger on this question. (I'm still trying to get hold of Bickel & Doksum.) $\endgroup$
    – kjo
    Commented Aug 30, 2017 at 0:55
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    $\begingroup$ Second edition of Casella & Berger, page 222. $\endgroup$
    – Zen
    Commented Aug 30, 2017 at 1:22
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    $\begingroup$ @Zen: Thank you! (I'm still shocked that neither "$t$-distribution" nor "Student" appears in such a book's index.) $\endgroup$
    – kjo
    Commented Aug 30, 2017 at 12:00

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