## Skin n' Bones CMS

Parag Magunia | April 17, 2021

Skin n’ Bones CMS is a hobby project of mine. I wasn’t satisfied with the journaling software currently available. Either the existing software wasn’t free or aesthetics of the journal wasn’t polished enough.

read more## Using AWS Rekognition To Turn Images and PDFs Into Human Readable Text

Parag Magunia | November 19, 2020

I wanted to turn a PDF into text I could copy and paste so I turned to AWS Rekognition. The console only accepted files that were 5 MB. Mine was 13 MB so I had to use the console with asynchronous methods.

read more## Installing Jekyll, OpenJDK 8 and S3 Website on a Mac

Parag Magunia | October 10, 2020

In this example we will install OpenJDK 8 and s3_website on a Mac with Catalina. It may work for other versions. This guide used to describe how to install Jekyll, but for that you should now visit the official Jekyll documentation at:

read more## The Real Number Axioms

Parag Magunia | October 4, 2020

The set of real numbers, $\mathbb{R}$, satisfies the following three sets of axioms.

read more## The Real Number System

Parag Magunia | October 4, 2020

The real number system, $\mathbb{R}$, is a ring, $R,$ definined on the quotient set of all rational Cauchy sequences such that the operations of addition and multiplication are defined as follows: $$[a_n] + [b_n] = [a_n + b_n] \\ [a_n][b_n] = [a_n b_n] $$

read more## Greek Letters

Parag Magunia | October 4, 2020

The upper and lowercase Greek letters used in Mathematics and Statistics.

read more## Types of Numbers

Parag Magunia | October 4, 2020

Common types of numbers found in Arithmetic. Integer any whole number including 0 such as 1, 2, or 3. $\frac{1}{2}$ is not a integer. $0.333\bar{3}$ is not an integer. $\frac{6}{3}$ is an integer even though it is a fraction because it reduces to 2 which is an integer. Negative whole numbers are also integers. The symbol for integers in mathematics is $\mathbb{Z}$ named after the German word 'zahlen.' Even Number Any number, $n$, that can be expressed as a multiple of 2. Now, 0 is an even number because $0 = 2 \cdot 0$. Negative numbers which are multiples of...

read more## Unary and Binary Ops

Parag Magunia | October 4, 2020

This page discusses unary and binary operations found in many Sets, Relations and Functions classes and also advanced Real Analysis. unary operation a mathematical function, say u, mapping a set S on to itself, i.e. $u:S \rightarrow S.$ binary operation a procedure carried out on two numbers. In particular, it is a function that maps the cartesian product of two elements in a set on to itself, i.e. $S \times S \rightarrow S.$ algebraic structure is a set, say S, along with a finite number of operations. Example: $ \{ \mathbb{Z^{+}},- \}.$ closed or closed under Let * be a...

read more## What is a fraction?

Parag Magunia | October 4, 2020

The word 'fraction' comes from the Latin 'fractus' which means 'broken.' A fraction is written as one number above another number with a line called the vinculum separating the two. The number on top is called the numerator which tells us the size of the quantity that is is going to be broken up. The denominator which is the number on the bottom tells us into how many pieces we are going to break up the numerator. All the divided pieces will be of the same size, and the size of the remaining pieces is what the fraction is equal...

read more## Converting Fractions

Parag Magunia | October 4, 2020

Suppose we had a fraction like $\frac12,$ and we wanted to convert it to decimaldecimal notation. All we have to do is divide 2 into 1 like this:

read more## Fractions to Decimals

Parag Magunia | October 4, 2020

Suppose we had a fraction like $\frac12,$ and we wanted to convert it to decimaldecimal notation. All we have to do is divide 2 into 1 like this:

read more## Binary Numeral System

Parag Magunia | October 4, 2020

A numeral system is a way of representing numbers, not to be confused with number systems which classify numbers based on properties.

read more## Binomial Distribution

Parag Magunia | October 4, 2020

In the last section, we took a first look at the some of the conditions necessary for a binomial experiment. Let's examine the binomial distribution more closely by examining the method to calculate the probability of getting a particular outcome for a series of bernoulli trials.

read more## Binomial Experiment

Parag Magunia | October 4, 2020

The binomial distribution is a discrete distribution which implies its sample space is finite. In other words, the values that a binomial random variable can take on is limited. The mean and variance is given by $ \mu = np $ and $ \sigma^2 = npq,$ respectively.

read more## Binomial Testing II

Parag Magunia | October 4, 2020

Suppose we conducted our experiment from the last section, and we got three heads. That is getting 2 fewer heads than would be expected by chance alone if the coin is indeed fair. Now we need to calculate the p-value of the experiment. The p-value is the probability of obtaining a test statistic at least as extreme as the observed value. In this case, our observed value is three. More extreme cases would be if we flipped the coin and we get 0, 1, or 2 heads. However, this is a two-sided test so we can get extreme cases on...

read more## Binomial Testing

Parag Magunia | October 4, 2020

In the last experiment, we assumed we had a fair coin. Suppose we wanted to conduct an experiment to see whether it was indeed a fair coin. First we need to state the null and alternative hypothesis and that is:

read more## Cardinality of Reals

Parag Magunia | October 4, 2020

Consider the cardinality of the real numbers. Here we will prove the real numbers are an infinite set that are uncountable. Let $f(x)=\frac{1}{1+n}| n \in \mathbb{N}.$ Here you can see that all the fractions in this sequence are between 0 and 1. It is also apparent that each fraction corresponds to a natural number. In that sense each natural number has been paired with a fraction. There are as many fractions as there are natural numbers. Yet this set of fractions is only a subset of the real numbers, therefore, the real numbers are not countably infinite. The real numbers...

read more## Cardinality

Parag Magunia | October 4, 2020

This page discusses cardinality, a basic topic found in classes like Sets, Relations and Functions and also Real Analysis. Definitions with discussion follow. numerically equivalent two sets are numerically equivalent if there exists a one-to-one function from one set onto the other. We can also say two sets have the same cardinal number if this is true. </dl> If we can say that a set has $n$ elements where $n$ is a specific element of the natural numbers, then we can say that particular set is countable. If each element in a infinite set can be paired with a natural...

read more## Cauchy Sequences

Parag Magunia | October 4, 2020

A sequence $a_n$ converges to $a$ if for every $\epsilon > 0$, $\exists$ an $N$ such that if $n > N$ then $\left| a_n - a \right| < \epsilon.$

read more## Decimal to Binary

Parag Magunia | October 4, 2020

Suppose we wanted to convert the decimal number 27 to binary. To get started we have to find out what place the first digit is going to be in. 27 is between $ 2^4=16 $ and $ 2^5=32 $. Our number is less than 32 so we can put a 0 in the "thirty-twos" place. We put a 0 in the thirty-twos place because if we put a 1 that would immediately turn the value of our binary number to at least 32. However, sixteen will fit inside 27 so we can put a 1 in the "sixteens" place.

read more## Fraction Operations I

Parag Magunia | October 4, 2020

The most basic fraction operations are addition and multiplication. So let's start there. There are two simple rules to follow when we add or multiply fractions.

read more## Fraction Terminology

Parag Magunia | October 4, 2020

factor a factor is an integer that divides another integer with no remainder leftover

read more## Hexadecimal Conversion

Parag Magunia | October 4, 2020

Hexadecimal is different from binary and decimal notation (base 10) in that there are sixteen unique digits whereas in decimal we have ten. Since we have sixteen digits we have to designate six new digits which are the letters A, B, C, D, E, and F. 'A' stands for 10, 'B' is 11, 'C' 12, 'D' 13, 'E' 14, and 'F' is 15.

read more## Logic and Quantifiers

Parag Magunia | October 4, 2020

This page discusses Logic and Quantifiers found in many Real Analysis and logic classes. Definitions with discussion follow. proposition an assertion that is either true or false but not both. A proposition is called simple or atomic if it is made up of only one proposition. A proposition is compound if it consists of one or more simple propositions with one or more logical connectives. logical connective joins simple propositions together to create a new proposition which may differ in truth value truth value the validity of a proposition. The truth value of a statement may be true or false...

read more## Mathematics Symbols

Parag Magunia | October 4, 2020

The most common symbols found in mathematics are listed below. The table is not exhaustive but covers numerals, trigonometry and geometry. If you would like to see a mathematics symbol added you can contact North Penn Networks. Likewise if there are errors please also reach out. The math symbols below are rendered with MathJax. $+$plus or additon sign $-$minus or subtraction sign $\times$times or multiplication sign $\div$division sign $=$equal sign $\neq$inequality $\gt$greater than $\geq$greater than or equal to $\lt$less than $\leq$less than or equal to $\%$percent $\sqrt{n}$square root of n $\left|n\right|$absolute value of n $\infty$infinity $\parallel$parallel $\perp$perpendicular $\angle$angle $!$factorial $!!$double...

read more## Moments

Parag Magunia | October 4, 2020

The mean is an another name for the average. You can think of the mean as a typical representative value of all the observations in a set. The standard deviation is a measure of the “spread” of the data. For example, if in a set of observations, the data was not concentrated around the mean, then that dataset would have a high standard deviation. If all the data, was concentrated around some single value, then that dataset would have a low standard deviation. The standard deviation squared is the variance. Another name for these measurements are moments.

read more## Number Systems

Parag Magunia | October 4, 2020

This page discusses the types of number systems found in real analysis and introduction to proving classes. Number System is a set of numbers together with one or more operations. Symbol Name Set $\mathbb{N}$ Natural Numbers {1, 2, 3, ...} $\mathbb{Z}$ Integers {...,-3, -2, -1, 0, 1, 2, 3, ...} $\mathbb{Q}$ Rational Numbers $\{\frac{a}{b}| a,b \in \mathbb{Z}\}$. All fractions. $\mathbb{R} - \mathbb{Q}$ Irrational Numbers Numbers which cannot be expressed in the form of a fraction. $\mathbb{R}$ Real Numbers Satisfies the Real Number Axioms. All finite and infinite decimal numbers. $\mathbb{C}$ Complex Numbers $\{a+bi \ \big| \ a,b \in \mathbb{R} \text{...

read more## Number Theory

Parag Magunia | October 4, 2020

Factor to factor means to write as a product Factor 6 $ 6 = 2 \cdot 3 $ When factoring, it is often useful to list only the prime factors. Usually, we do not list 1 as a factor. Prime A prime number is a natural number (positive, whole) only divisible by 1 and itself. A prime number must also be greater than 1. 2 is the first prime number. 2 is the only even prime number. Composite A composite number is a natural number greater than 1 that is not prime. Factor 26 $ 2 \cdot 13 $ Factor...

read more## Ordered Fields

Parag Magunia | October 4, 2020

A ring is a set $R$ with the usual operations of multiplication and addition from $R \times R \rightarrow R$ which satisfy the properties of: associativity under addition commutativity under addition the existence of additive identity the existence of an additive inverse associativity under multiplication distributivity across addition These properties are defined in the above mentioned link to the page real number axioms.

read more## Partial Orders

Parag Magunia | October 4, 2020

A partial order is a relation that satisfies three requirments:

read more## Random Variables

Parag Magunia | October 4, 2020

A random variable, $ X $, is a function that assigns to each element in the sample space a unique value. The sample space is the domain of $X$. The value assigned to each element in the sample space is the probability for that particular element in the sample space occurring. The sum of the probabilities over the sample space must equal 1, and the probability that the random variable takes on a value $ x $ must be between 0 and 1, inclusive.

read more## Sets, Relations and Funcs

Parag Magunia | October 4, 2020

This page discusses Sets,Relations and Function. The ideas presented here only skim the surface. Definitions and discussion follow. relation A subset $R$ of $A \times B$ is a relation from $A$ to $B$. inverse If $R$ is $\{(a, b) | (a,b) \in R\}$, then $R^{-1} = \{(b, a) | (a, b) \in R\}$ function a mapping of $f$ from $X$ into $Y$ such that if $(x,y_1) \in f$ and $(x,y_2) \in f$, then $y_1=y_2$. To designate this we write $f:X \rightarrow Y$. onto if f is a function from $X$ to $Y,$ and the range(see below) of $f$ is $Y$...

read more## Sets

Parag Magunia | October 4, 2020

This page discusses sets, a topic that forms the foundation of arithmetic and real analysis alike. set a collection element a member of a set. We denote $x$ is a element of set $S$ by $x \in S$ empty set, $\varnothing$ the set with no elements subset a set $S$ is a subset of $T$, written $S \subset T$, if $ \forall x$ if $ x \in S \Rightarrow x \in T $ finite set a set in which each element can be matched with exactly one element of the set $\{1, 2, ..., n\}$ set equality two sets $S$...

read more## Statistical Populations

Parag Magunia | October 4, 2020

In statistics, we are often interested in the population which is some entire set of entities that share a common trait. We often draw a sample which is a subset of the population to draw inferences about the population. The set of measurements taken from the sample form the sample data.

read more## Subtraction and Division

Parag Magunia | October 4, 2020

The most common terms found in subtraction and division arithmetic are listed below. If you find any math errors, you may reach out to North Penn Networks using the contact form. $ a-b=c $ minuend the first number in a subtraction operation; the subtrahend is taking away from this number; the minuend is $a$ subtrahend the second number in a subtraction operation; the amount taken away from the minuend; the subtrahend is $b$ difference the result of taking the subtrahend from the minuend; the difference is $c$ $ a \div b = c $ dividend the number being divided, $a$...

read more## The Decimal System

Parag Magunia | October 4, 2020

The decimal system is a way of representing a number using powers of ten. The first digit before the decimal is the ones place. Suppose we have a digit, $ d $, that is in the ones place. Since $ d $ is in the ones place the value of $ d = d \times 10^0. $ We use 10 since we are in the decimal system (base 10). We use 0 for the exponent because we are 1 digit to the left of the decimal. Why 0? In Computer Science and Mathematics, counting is often started at 0. Also...

read more## Addition Tables

Parag Magunia | September 28, 2020

The idea of numerals is illustrated below. The numbers from zero to nine are depicted as dominoes shaded in blue. Though the numbers are depicted below, numbers are imaginary (they don't exist in real life). I've never seen the number five walking in my neighborhood or any other number for that fact. In this respect math is different from other sciences than say, chemistry, where atoms can be reached out and touched or geology where the Earth can be walked upon. After the "imaginary" numerals depicted below some addition and multiplication terminology is defined in the context of real analysis....

read more## North Penn Networks

Parag Magunia | January 1, 2020

2019 was a very good year thanks to two contracts that took place at the beginning and end of the year. Both were related to Drupal with some hints of AWS. I got to configure load balancers and also a large RDS MySQL instance. I signed non-disclosure agreements with my client's so I'm not able to mention their name though one of them was a large food service company based out of the midwest US and the other was for a travel sight that sees a lot of visitors. As I'm currently a student at the University of Illinois at...

read more## Illinois Beginnings

Parag Magunia | July 6, 2019

UPDATE - January 3, 2020 Just got back my grade for the first semester and I completed the Cloud Networking course successfully with full credit (4 hours). My GPA is a 3.67. Looking forward to next semester's Cloud Computing Applications course. The semester starts January 21, 2020. There will be a final and mid-term. Good luck Parag! I recently got admitted to the University of Illinois at Urbana-Champaign. Their MCS (Master of Computer Science) program has strong data science requirements. I'm looking forward to the Cloud Computing courses offered as part of the degree requirements starting this Fall with Cloud...

read more## Picostat Endings

Parag Magunia | July 18, 2018

I used to have a number of R Datasets available for download and visual inspection. They were cloned from the datasets found in R, and I created Slickgrid-enabled webpages out of the data for easy inspection. I also added support for dataset management and basic statistical operations. At one point there were 3,000 visitors to the site every month.

read more## Terms of Use

North Penn Networks Limited | January 1, 2017

Disclaimer of Warranty THERE IS NO WARRANTY FOR THE SITE OR ITS SERVICES, TO THE EXTENT PERMITTED BY APPLICABLE LAW. THE SITE OWNERS, SERVICE PROVIDERS, COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE SITE AND SERVICES "AS IS" WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE SITE AND SERVICES IS WITH YOU. SHOULD THE SITE OR SERVICES PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION. IN NO EVENT...

read more## Design Credits

Parag Magunia | January 1, 2017

The Prologue theme for this website was originally designed by HTML5 Up. Chris Bobbe adapted the theme for Jekyll. Both themes are licensed under Creative Commons Attribution 3.0 Unported. Modifications of the Jekyll theme were made by Parag Magunia.

read more## Privacy Policy

Parag Magunia | January 1, 2017

North Penn Networks Limited, hereafter known as I, or me, has a strong commitment to privacy on the Internet. I want my site, Picostat.com, to be useful and informative to you. You may thus have the opportunity to provide personal information and data through my site. This Statement sets forth my privacy policy for this website and describes the procedures I follow to respect the privacy of visitors to this site and their information. User-created datasets, analyses, comments, and documentation created on this site automatically authorize me to host copies on my servers though copyright shall remain with the user....

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