# Elements of Probability Part 1

Its 6:00 AM, a beautiful day. The air is crisp, but not cold. There is a freshness that remains from the morning dew, and the smile of the sun is breaking the horizon. The door closes, your key turns the ignition switch and the current moves to your starter -but, unfortunately, the load does not… Read More Elements of Probability Part 1

# Linear Model Considerations

In our post on LSR, we discussed some of the basic formulas and methods to derive a linear model. Today, our goal will be to quickly review multiple linear regression, and evaluate how we interpret linear regression models. Simple linear regression is useful, but what about situations where one has to model a significant number… Read More Linear Model Considerations

# Simple Linear Regression

Want a straightforward approach to prediction? Nothing is as direct as simple linear regression (SLR). Suppose that you want to predict a quantitative response variable on the basis of a single  response variable . It assumes a linear relationship amid the two variables. We say: [1] We can think of (1) as the regression of on -this… Read More Simple Linear Regression

# Lebesgue Theory Part 3

Previously, we defined a measure m*(I) to be the infimum of the sum of the components (covers) of the measure (i.e. due to the measure being additive). We talked about moving forward and refining our example. In some texts, they continue with denoting a measure using m(I). However, most texts move forward with a different… Read More Lebesgue Theory Part 3

# Lebesgue Theory Part 2

Part 1 of this post focused on an introduction into sets, what a ring is, what a sigma-ring is, what does it mean to be additive, independent, and countably additive. We finished with a theorem that addressed the essence of countable additivity; this theorem set up the needed tools to evaluate some important aspect of… Read More Lebesgue Theory Part 2

# Lebesgue Theory Part 1 (Set Functions)

What is a set function? In fact, what is a function? Before jumping into the deep water, let us take a moment to review the idea of a function, what it is, and how it is useful to us. 1) Definition: Given two non-empty sets and a function  is a rule (i.e. a set of instructions) where each… Read More Lebesgue Theory Part 1 (Set Functions)

# Pointwise Limits, Uniform Limits, and Continuity

Before diving into the muck head first, take a moment to go over some of the basic ideas involving continuity -particularly, uniform continuity. We will lightly cover a few concepts of continuity to provide a base for our discussion. We know from basic calculus and our study of integration, differentiation, and continuity that: on an… Read More Pointwise Limits, Uniform Limits, and Continuity

# Monotonicity and Derivatives

Now, we learn from calculus that the slope of a function can be found by way of its derivative; what is more fascinating is that we can go further -much further. Definition #1: Let be real valued on an interval . The following holds: 1) If whenever and are in with  , then our function is nondecreasing on… Read More Monotonicity and Derivatives

# Cauchy

Open any advanced calculus text or work in any field of analysis, you are likely to relish in the glorious ideas of Agustin-Louis Cauchy and not even know it. Cauchy was born in Paris, France in 1789. He began his professional life as an engineer and progressively influenced mathematics by way of research, professorships (when… Read More Cauchy

# The Baire Category Theorem

I doubt many of you are familiar with the Baire Category Theorem (BCT) -on second thought, you might; I mean, you have taken the time to read this article (I am guessing that only a subset of a certain group of people read random math/philosophy blogs in their spare time).  I have been involved in… Read More The Baire Category Theorem
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://problemofinduction.com/category/mathematics/", "fetch_time": 1675057723000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2023-06/segments/1674764499801.40/warc/CC-MAIN-20230130034805-20230130064805-00137.warc.gz", "warc_record_offset": 476965505, "warc_record_length": 19123, "token_count": 891, "char_count": 3875, "score": 3.75, "int_score": 4, "crawl": "CC-MAIN-2023-06", "snapshot_type": "latest", "language": "en", "language_score": 0.8975409269332886, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00053-of-00064.parquet"}