By Joseph K. Blitzstein, Jessica Hwang

ISBN-10: 1466575573

ISBN-13: 9781466575578

Constructed from celebrated Harvard information lectures, creation to chance presents crucial language and instruments for figuring out facts, randomness, and uncertainty. The e-book explores a wide selection of purposes and examples, starting from coincidences and paradoxes to Google PageRank and Markov chain Monte Carlo (MCMC). extra software components explored comprise genetics, medication, machine technological know-how, and knowledge concept.

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100 2 ). In math, v + w is undefined if v and w are vectors of di↵erent lengths, but in R the shorter vector gets “recycled”! For example, v+3 adds 3 to each entry of v. Factorials and binomial coefficients We can compute n! using factorial(n) and nk using choose(n,k). As we have seen, factorials grow extremely quickly. What is the largest n for which R returns a number for factorial(n)? Beyond that point, R will return Inf (infinity), with a warning message. ). Similarly, lchoose(n,k) computes log nk .

4 (Partnerships). Let’s use a story proof to show that (2n)! = (2n 2n · n! 1)(2n 3) · · · 3 · 1. Story proof : We will show that both sides count the number of ways to break 2n people into n partnerships. Take 2n people, and give them ID numbers from 1 to 2n. We can form partnerships by lining up the people in some order and then saying the first two are a pair, the next two are a pair, etc. This overcounts by a factor of n! · 2n since the order of pairs doesn’t matter, nor does the order within each pair.

A) Give an example of independent events A and B in a finite sample space S (with neither equal to ; or S), and illustrate it with a Pebble World diagram. (b) Consider the experiment of picking a random point in the rectangle R = {(x, y) : 0 < x < 1, 0 < y < 1}, where the probability of the point being in any particular region contained within R is the area of that region. Let A1 and B1 be rectangles contained within R, with areas not equal to 0 or 1. Let A be the event that the random point is in A1 , and B be the event that the random point is in B1 .

### Introduction to Probability by Joseph K. Blitzstein, Jessica Hwang

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