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Calculus and Statistics (Dover Books on Mathematics)

Michael C. Gemignani · 2 HN comments
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Amazon Summary
Self-contained and suitable for undergraduate students, this text offers a working knowledge of calculus and statistics. It assumes only a familiarity with basic analytic geometry, presenting a coordinated study that develops the interrelationships between calculus, probability, and statistics. Starting with the basic concepts of function and probability, the text addresses some specific probabilities and proceeds to surveys of random variables and graphs, the derivative, applications of the derivative, sequences and series, and integration. Additional topics include the integral and continuous variates, some basic discrete distributions, as well as other important distributions, hypothesis testing, functions of several variables, and regression and correlation. The text concludes with an appendix, answers to selected exercises, a general index, and an index of symbols.
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There is SICP.

For distributed systems I'd read Lynch's book on Distributed Algorithms.

For stats Michael Freedman's book, Statistics, is a good simple introduction. Someone mentioned Calculus and Statistics (http://www.amazon.com/Calculus-Statistics-Dover-Books-Mathem...) here on HN a couple of weeks ago. I had looked it when it came out and re-reviewed it with that thread -- I really like it a fair bit more than I remember and a better text for those looking for a more rigorous treatment than Freedman -- although probably still too simple if you're reading Foundations of Statistical NLP.

There's also Scott's book on Programming Languages, which is worth reading.

> wish schools taught math leading to statistics and probability instead of leading to calculus

This is silly. All probability distributions are cadlag, so how can you even teach probability without the notion of right continous with left limits, which means you have to resort to limits & derivatives => Calc.

Actually, the argument for combining Calc & Stats is very compelling, because there is too much synergy. How can you teach a continous probability distribution like say the Gaussian without teaching how to integrate under the curve for the cumulative distribution function, or obtaing the probability density function via the derivative, or obtaining the variance aka second central moment via the moment generating function, which means you now have to teach atleast some fourier transforms which again means Calculus. At both UChicago & Stanford where I learnt all of my probability, calculus was quite intertwined with the teaching of probability. I believe its the same case in most other schools as well.

Without calc in probability, you can do "lame" stuff like discrete distributions ( Binomial, Poisson etc....but even there, the key insight is to show how the CDFs of the discrete distributions, which will generally have terribly complicated formulae with giant factorial expressions, can be very nicely approximated by the continous distributions for large n, small p etc. ( aka continous correction http://en.wikipedia.org/wiki/Continuity_correction ). So for a large number of coin flips trials, you use a Normal to approximate the CDF because otherwise the original binomial CDF is too hard to compute with your TI-84s (because you have one giant factorial divided by another giant factorial and the numerical overflows will kill the computation unless you are very careful about how you go about computing the result).

My favorite go-to guide remains the excellent Calc & Stat Dover book ( http://www.amazon.com/Calculus-Statistics-Dover-Books-Mathem... ), which combines Calc & Stats from page 1. There is simply no better way to learn stats than via calc.

timwiseman
You are right in the sense you absolutely cannot get a deep understanding of Statistics without Calculus.

But with a mere background of high school algebra, you can learn more about Stats than most college graduates have, and that knowledge is far more relevant to the day-to-day lives of the average person in America than Calculus is.

comicjk
True, because the average person in America has a pretty crappy job.

Those who get a chance to follow through with calculus and apply it do much, much better.

kenjackson
You're talking college stats and calc. And even then Chicago and Stanford. It's like saying that Harvard's Math 55 should be the model for intro math courses in college.

I think the original poster was thinking high school and a target much more like Freedman's text (http://www.amazon.com/Statistics-4th-David-Freedman/dp/03939...).

And what Freedman's book does probably better than any other text in the field is teach how to think about statistics. It doesn't have a lot of formalisms, but if can come to an understanding of what he teaches in that book you'll have a rich understanding of stats.

With that said, if Calc was taught in the context of functions and probability, as in the Gemignani text then I think we'd be better off than how Calc today is focused around engineering.

Sukotto
I'm talking at a simpler level than you are.

From what I can tell (and remember), elementary & high school math is specifically designed to take you from 0 to calculus. (well, maybe not ALL the way unless you take AP math)

Personally, I find basic stats and prob far more valuable in day to day life than calculus. So my point was just that I wish schools would focus on that area of math as the goal.

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