Fixed titles, math rendering, and links on some pages

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Brandon Rozek 2021-07-26 09:13:20 -04:00
parent 9f096a8720
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61 changed files with 303 additions and 115 deletions

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# Abstract Algebra 2 Definitions
---
title: Abstract Algebra Notes
showthedate: false
math: true
---
Chapter markings are based off the book "A Book of Abstract Algebra" by Charles C. Pinter.
## Chapter 17

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title: Algorithms Book Study
---
# Algorithms Book Study
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/)!
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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[Chapter 3](dynamic)
[Chapter 4](greedy)
[Chapter 4](greedy)

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# Backtracking
---
title: Backtracking
showthedate: false
---
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*.
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.
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
---
title: Dynamic Programming
---
The book first goes into talking about the complexity of the Fibonacci algorithm
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## Greedy Algorithms
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.
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
---
title: Greedy Algorithms
showthedate: false
---
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
1. An arbitrary unmatched hospital A offers its position to the best doctor a who has not already rejected it.
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.
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
---
title: Recursion
showthedate: false
math: true
---
## Reductions

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---
title: Bayesian Statistics
title: Bayesian Statistics - From Concept to Data Analysis
showthedate: false
---
# Bayesian Statistics: From Concept to Data Analysis
In the Winter of 2017, I took a course on Bayesian Statistics on Coursera offered by Dr. Herbert Lee.
Below are the notes for each of the four weeks.

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# Bayesian Statistics
---
title: Week 1
showthedate: false
math: true
---
## Rules of Probability

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---
title: Week 2
showthedate: false
math: true
---
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.
## 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."
Bayesian credible intervals have the interpretation that "Your posterior probability that the parameter is in a 95% credible interval is 95%."
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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---
title: Week 3
showthedate: false
math: true
---
How do we choose a prior?
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.
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.
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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---
title: Week 4
showthedate: false
math: true
---
## Exponential Data
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
---
title: Handy Facts about Quadratic Congruences
showthedate: false
math: true
---
## Number of Solutions

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# Real Analysis Sheet
---
title: Real Analysis Quick Sheet
showthedate: false
math: true
---
**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
---
title: Week 1
showthedate: false
---
## Replication
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It should not include every analysis you performed
References should be included for statistical methods
References should be included for statistical methods

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---
title: Week 2
showthedate: false
---
## Coding Standards for R
1. Always use text files/text editor

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---
title: Week 3
showthedate: false
---
## tl;dr
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
- Evidence-based data analysis would provide standardized best practices for given scientific areas and questions
- Gives reviewers an important tool without dramatically increases the burden on them
- More effort should be put into improving the quality of "upstream" aspects of scientific research
- More effort should be put into improving the quality of "upstream" aspects of scientific research

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---
title: Week 4
showthedate: false
---
## The `cacher` Package for R
- Add-on package for R

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showthedate: false
---
# Probability and Statistical Inference
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.
*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
**Brandon Rozek**
---
title: Central Limit Theorem
showthedate: false
math: true
---
## Introduction

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# Confidence Interval Lab
**Written by Brandon Rozek**
---
title: Confidence Interval
showthedate: false
---
## Introduction

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# Random Number Generation
---
title: Random Number Generation
showthedate: false
math: true
---
## Introduction

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# Random Walk
---
title: Random Walk
showthedate: false
math: true
---
## Introduction