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Fixed titles, math rendering, and links on some pages
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# Abstract Algebra 2 Definitions
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---
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title: Abstract Algebra Notes
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showthedate: false
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math: true
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---
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Chapter markings are based off the book "A Book of Abstract Algebra" by Charles C. Pinter.
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## Chapter 17
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title: Algorithms Book Study
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---
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# Algorithms Book Study
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A couple of my friends and I decided to start a book club following "Algorithms" by Jeff Erickson. One bonus is that he gives it away for free on [his website](http://jeffe.cs.illinois.edu/teaching/algorithms/)!
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Of course you should totally check his book out rather than reading my notes. There are tons of witty and fun things in his textbook, not a dry reading I promise. These notes are here mostly for archival purposes.
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@ -14,4 +13,4 @@ Of course you should totally check his book out rather than reading my notes. Th
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[Chapter 3](dynamic)
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[Chapter 4](greedy)
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[Chapter 4](greedy)
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# Backtracking
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---
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title: Backtracking
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showthedate: false
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---
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This algorithm tries to construct a solution to a problem one piece at a time. Whenever the algorithm needs to decide between multiple alternatives to the part of the solution it *recursively* evaluates every option and chooses the best one.
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Backtracking algorithms are used to make a *sequence of decisions*.
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When we design a new recursive backtracking algorithm, we must figure out in advance what information we will need about past decisions in the middle of the algorithm.
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When we design a new recursive backtracking algorithm, we must figure out in advance what information we will need about past decisions in the middle of the algorithm.
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# Dynamic Programming
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---
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title: Dynamic Programming
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---
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The book first goes into talking about the complexity of the Fibonacci algorithm
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## Greedy Algorithms
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If we're lucky we can just make decisions directly instead of solving any recursive subproblems. The problem is that greedly algorithms almost never work.
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If we're lucky we can just make decisions directly instead of solving any recursive subproblems. The problem is that greedly algorithms almost never work.
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# Greedy Algorithms
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---
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title: Greedy Algorithms
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showthedate: false
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---
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Greedy Algorithms are about making the best local choice and then blindly plowing ahead.
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The Gale-Shapley algorithm is a great greedy fit. It goes like this
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1. An arbitrary unmatched hospital A offers its position to the best doctor a who has not already rejected it.
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2. If a is unmatched, she tentatively accepts A's offer. If a already had a match but prefers A, she rejects her current match and tentatively accepts the new offer from A. Otherwise a rejects the new offer.
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2. If a is unmatched, she tentatively accepts A's offer. If a already had a match but prefers A, she rejects her current match and tentatively accepts the new offer from A. Otherwise a rejects the new offer.
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# Recursion
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---
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title: Recursion
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showthedate: false
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math: true
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---
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## Reductions
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---
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title: Bayesian Statistics
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title: Bayesian Statistics - From Concept to Data Analysis
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showthedate: false
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---
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# Bayesian Statistics: From Concept to Data Analysis
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In the Winter of 2017, I took a course on Bayesian Statistics on Coursera offered by Dr. Herbert Lee.
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Below are the notes for each of the four weeks.
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# Bayesian Statistics
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---
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title: Week 1
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showthedate: false
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math: true
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---
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## Rules of Probability
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---
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title: Week 2
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showthedate: false
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math: true
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---
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Under the frequentest paradigm, you view the data as a random sample from some larger, potentially hypothetical population. We can then make probability statements i.e, long-run frequency statements based on this larger population.
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## Coin Flip Example (Central Limit Theorem)
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Frequentest confidence intervals have the interpretation that "If you were to repeat many times the process of collecting data and computing a 95% confidence interval, then on average about 95% of those intervals would contain the true parameter value; however, once you observe data and compute an interval the true value is either in the interval or it is not, but you can't tell which."
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Bayesian credible intervals have the interpretation that "Your posterior probability that the parameter is in a 95% credible interval is 95%."
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Bayesian credible intervals have the interpretation that "Your posterior probability that the parameter is in a 95% credible interval is 95%."
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---
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title: Week 3
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showthedate: false
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math: true
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---
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How do we choose a prior?
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Our prior needs to represent our personal perspective, beliefs, and our uncertainties.
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2. In Bayesian Statistics, a vague prior refers to one that's relatively flat across much of the space. For a Gamma prior we can choose $\Gamma(\epsilon, \epsilon)$ where $\epsilon$ is small and strictly positive.
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This would create a distribution with a mean of 1 and a huge standard deviation across the whole space. Hence the posterior will be largely driven by the data and very little by the prior.
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This would create a distribution with a mean of 1 and a huge standard deviation across the whole space. Hence the posterior will be largely driven by the data and very little by the prior.
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---
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title: Week 4
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showthedate: false
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math: true
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---
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## Exponential Data
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Suppose you're waiting for a bus that you think comes on average once every 10 minutes, but you're not sure exactly how often it comes.
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# Handy Quadratic Congruences Facts
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---
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title: Handy Facts about Quadratic Congruences
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showthedate: false
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math: true
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---
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## Number of Solutions
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# Real Analysis Sheet
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---
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title: Real Analysis Quick Sheet
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showthedate: false
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math: true
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---
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**Fact:** $\forall a,b, \in \mathbb{R}$, $\sqrt{ab} \le \frac{1}{2}(a + b)$.
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(b) If ($r = 0$ and $\sum{y_n} < \infty$), then $\sum{x_n} < \infty$.
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# Reproducible Research Week 1
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---
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title: Week 1
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showthedate: false
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---
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## Replication
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It should not include every analysis you performed
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References should be included for statistical methods
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References should be included for statistical methods
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---
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title: Week 2
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showthedate: false
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---
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## Coding Standards for R
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1. Always use text files/text editor
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---
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title: Week 3
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showthedate: false
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---
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## tl;dr
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People are busy, especially managers and leaders. Results of data analyses are sometimes presented in oral form, but often the first cut is presented via email.
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- Reproducible research focuses on the most "downstream" aspect of research documentation
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- Evidence-based data analysis would provide standardized best practices for given scientific areas and questions
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- Gives reviewers an important tool without dramatically increases the burden on them
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- More effort should be put into improving the quality of "upstream" aspects of scientific research
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- More effort should be put into improving the quality of "upstream" aspects of scientific research
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---
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title: Week 4
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showthedate: false
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---
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## The `cacher` Package for R
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- Add-on package for R
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showthedate: false
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---
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# Probability and Statistical Inference
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In the Fall of 2017, I took the course STAT 381 with Dr. Debra Hydorn. Below I included the interesting labs we worked on in the class.
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*Please note that these reports were not formatted for this site. So equations and images may not show up.*
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# Central Limit Theorem Lab
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**Brandon Rozek**
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---
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title: Central Limit Theorem
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showthedate: false
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math: true
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---
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## Introduction
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# Confidence Interval Lab
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**Written by Brandon Rozek**
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---
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title: Confidence Interval
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showthedate: false
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---
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## Introduction
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# Random Number Generation
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---
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title: Random Number Generation
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showthedate: false
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math: true
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---
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## Introduction
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# Random Walk
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---
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title: Random Walk
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showthedate: false
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math: true
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---
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## Introduction
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