MS1023 Marginal vs Conditional Probability Matrix

19 MOOCs on Mathematics & Statistics for Data Science & Machine Learning

**original source : https://www.analyticsvidhya.com/blog/2017/01/19-mooc-mathematics-statistics-datascience-machine-learning/**

## Introduction

Before creation, God did just pure mathematics. Then he thought it would be pleasant change to do some applied

-John Edensor Littlewood

Mathematics & Statistics are the founding steps for data science and machine learning. Most of the successful data scientists I know of, come from one of these areas – computer science, applied mathematics & statistics or economics. If you wish to excel in data science, you must have a good understanding of basic algebra and statistics.

However, learning Maths for people not having background in mathematics can be intimidating. First, you have to identify what to study and what not. The list can include Linear Algebra, calculus, probability, statistics, discrete mathematics, regression, optimization and many more topics. What do you do? How deep to you want to get in each of these topics? It is very difficult to navigate through this by yourself.

If you have faced this situation before – don’t worry! You are at the right place now. I have done the hard work for you. Here is a list of popular open courses on Maths for Data science from Coursera, edX, Udemy and Udacity. The list has been carefully curated to give you a structured path to teach you the required concepts of mathematics used in data science.

Get started now to learn & explore mathematics for data science.

## Which course is suitable for you?

To help you navigate through the courses, I have divided the article into beginners, intermediate and advanced section. Choose your level of expertise in mathematics before delving further. Further, I have added the pre-requisites for each course. You can check if you know these topics before starting the course.

Few courses may require you to finish the preceding course for better understanding. So, make sure that you either know the subject or have undergone these courses.

Read on to find out the right course for you!

## Table of Content

- Beginners Mathematics / Statistics
- Intermediate Mathematics / Statistics
- Advanced Mathematics / Statistics

- Data Science Maths Skills
- Intro to Descriptive Statistics
- Intro to Inferential Statistics
- Introduction to Probability and Data
- Math is Everywhere: Applications of Finite Math
- Probability: Basic Concepts & Discrete Random Variables
- Mathematical Biostatistics Boot Camp 1
- Applications of Linear Algebra Part 1
- Introduction to Mathematical Thinking

- Bayesian Statistics: From Concept to Data Analysis
- Game Theory 1
- Game Theory II: Advanced Applications
- Advanced Linear Models for Data Science 1: Least Squares
- Advanced Linear Models for Data Science 2: Statistical Linear Models
- Introduction to Linear Models and Matrix Algebra
- Maths in Sports

- Discrete Optimization
- Statistics for Genomic Data Science
- Biostatistics for Big Data Applications

## Beginners Mathematics & Statistics

1.

Duration: 4 weeks

Led by: Duke University (Coursera)

If you are a beginner with very minimal knowledge of mathematics, then this course is for you. In this course, you will learn about concepts of algebra like set theory, inequalities, functions, coordinate geometry, logarithms, probability theory and many more.

This course will take you through all the basic maths skills required for data science and would provide a strong foundation.

The course starts from 9 Jan 2017 and is lead by professors from Duke University.

Prerequisites: Basic maths skills

2.

Intro to Descriptive Statistics

Duration: 8 weeks

Led by: Udacity

This course by Udacity is an excellent beginners guide for learning statistics. It is fun, practical and filled with examples. The Descriptive Statistics course will first make you familiar with different terms of statistics and their definition. Then you will learn about statistics concepts like central tendency, variability, standard normal distribution and sampling distribution.

This course doesn’t require any prior knowledge of statistics and is open for enrollment.

Prerequisites: None

3.

Intro to Inferential Statistics

Duration: 8 weeks

Led by: Udacity

After you have gone through the Descriptive Statistics course, it is time for Inferential statistics. The same practical approach to the subject continues in this course.

In this course, you will learn concepts of statistics like estimation, hypothesis testing, t-test, chi-square test, one-way Anova, two-way Anova, correlation, and regression.

There are problem set and quiz questions after each topic. You will also be able to test your learning on a real-life dataset at the end of the course. The course is open for enrollment.

Prerequisites: Complete understanding of Descriptive Statistics (the course mentioned above)

Alternate Course: You can also look at Statistics: Unlocking the World of Data. It is a 6 weeks long course run by University of Edinburgh (edX)

4.

Introduction to Probability and Data

Duration: 5 weeks

Led by: Duke University (Coursera)

It will provide you hands on experience in data visualization and numeric statistics using R and RStudio.

The course will first take you through basics of probability and data exploration to give a basic understanding to get started. Then, it will individually explain various concepts under each topic in detail. At the end, you will be tested on a data analysis project using a real-world dataset.

The course is led by a Professor in Statistics at Duke University and is also a prerequisite for Statistics in R specialization. If you are looking forward to learn R for data science, then you must take this course. The course is open for enrollment.

Prerequisites: Basic Statistics and knowledge of R

5.

Math is Everywhere: Applications of Finite Math

Duration: 1 week

Led by: Davidson College (Udemy)

As the name suggests, this course tells you how maths is being used everywhere from Angry birds to Google. It is a fun approach to applied mathematical concepts.

In this course, you will learn how equation of lines is used to create computer fonts, how graph theory plays a vital role in angry birds, linear systems model the performance of a sports team and how Google uses probability and simulation to lead the race in search engines.

The course is led by the mathematics professor at Davidson College and is open for enrollment.

Prerequisites: Understanding of linear algebra and programming

6.

Probability: Basic Concepts & Discrete Random Variables

Duration: 6 weeks

Led by: Purdue University (edX)

This course is designed for anyone looking for a career in data science & information science. It covers essentials of mathematical probabilities.

In this course, you will learn the basic concepts of probability, random variables, distributions, Bayes Theorem, probability mass functions and CDFs, joint distributions and expected values.

Once you are familiar with the basics, you will learn about advanced concepts Bernoulli and Binomial distributions, Geometric distribution, Negative Binomial distribution, Poisson distribution, Hypergeometric distribution and discrete uniform distribution.

After taking this course you will have a thorough understanding of how probability is used in everyday life. The course is open for enrollment.

Prerequisite: Basics Statistics

7.

Mathematical Biostatistics Boot Camp 1

Duration: 4 weeks

Led by: Johns Hopkins University (Coursera)

Honestly, the “Bio” in “Biostatistics” is misleading. This course is all about fundamental probability and statistics techniques for data analysis.

The course covers topics on probability, expectations, conditional probabilities, distributions, confidence intervals, bootstrapping, binomial proportions, and logs.

A prior knowledge of linear algebra and programming will be advantageous but not mandatory to begin with this course. The course starts from 16 Jan 2017 and is led by biostatistics professor at Johns Hopkins University.

A well-paced course with a complete introduction to mathematical statistics.

Prerequisites: Basic Linear algebra, calculus and programming useful but not mandatory

8.

Applications of Linear Algebra Part 1

Duration: 5 weeks

Led by: Davidson College (edX)

This is an interesting course on applications of linear algebra in data science.

The course will first take you through fundamentals of linear algebra. Then, it will introduce you to applications of linear algebra for recognizing handwritten numbers, ranking of sports team along with online codes.

The course is open for enrollment.

Prerequisite: Basic linear algebra

9.

Introduction to Mathematical Thinking

Duration: 8 weeks

Led by: Stanford University (Coursera)

In this mathematical thinking course from Stanford, you will learn how to develop analytical thinking skills. The course teaches you interesting ways to develop out-of-the-box thinking and helps you remain ahead of the competitive curve.

In this course, you will learn about analysis of a language, quantifiers, brief introduction to number theory and real analysis. To make the most of this course one must have familiarity with algebra, number system and elementary set theory.

The course starts from 9 Jan 2017 and is led by professors at Stanford. It is open for enrollment.

Prerequisites: Basic algebra, number system and elementary set theory.

## Intermediate Mathematics & Statistics

By this time, you know all the basic concepts a data scientist needs to know. This is the time to take your mathematical knowledge to the next level.

1.

Bayesian Statistics: From Concept to Data Analysis

Duration: 4 weeks

Led by: University of California (Coursera)

Bayesian Statistics is an important topic in data science. For some reason, it does not get as much attention.

In this course, the first section covers basic topics like probability like conditional probability, probability distribution and Bayes Theorem. Then you will learn about statistical inference for both Frequentist and Bayesian approach, methods for selecting prior distributions and models for discrete data and Bayesian analysis for continuous data.

Prior knowledge of statistics concepts is required to take this course. The course starts from 16 Jan 2017.

Prerequisite: Basic & Advanced Statistics

2.

Duration: 8 weeks

Led by: Stanford University and University of British Columbia (Coursera)

Game theory is an important component of data science. In this course, you will learn about basics of game theory and its applications. If you are looking to master Re-inforcement learning this year – this course is a must learn for you.

The course provides basic understanding of representing games and strategies, the extensive form (which computer scientists call game trees), Bayesian games (modeling things like auctions), repeated and stochastic games. Each concept has been explained with the help of examples and applications.

The course is led by professors from the Stanford University and The University of British Columbia. The course is open for enrollment.

Prerequisite: Basic probability and mathematical thinking

3.

Game Theory II: Advanced Applications

Duration: 5 weeks

Led by: Stanford University and The University of British Columbia (Coursera)

After going through the basics of Game theory in the previous course, this course is on the advanced applications of game theory.

You will learn about how to design interactions between agents in order to achieve good social outcomes. The three main topics covered are social choice theory, mechanism design, and auctions. The course starts from 30 Jan 2017 and is led by professors from Stanford University & The University of British Columbia.

The course is open for enrollment.

Prerequisite: Basics of Game Theory

4.

Introduction to Linear Models and Matrix Algebra

Duration: 4 weeks

Led By: Harvard University (edX)

Matrix algebra is used in various tools for experimental design and analysis of high-dimensional data.

For easy understanding, the course has been divided into seven parts to provide you a step by step approach. You will learn about matrix algebra notation & operations, application of matrix algebra to data analysis, linear models and QR decomposition.

The language used throughout the course is R. Feel free to choose which part of the course caters more to your interest and take the course accordingly.

The course is conducted by biostatistics professors at Harvard University and is open for enrolment now.

Prerequisite: Basic Linear algebra and knowledge of R

5.

Advanced Linear Models for Data Science 1: Least Squares

Duration: 6 weeks

Led by: Johns Hopkins University (Coursera)

This course is a two part series for advanced linear statistical learning models. For all those who have an understanding of regressions models and are looking to explore this topic further must take this course.

In this course, you will learn about one & two parameter regression, linear regression, general least square, least square examples, bases & residuals.

Before you proceed further let me clear, to take this course you need a basic understanding of linear algebra & multivariate calculus, statistics & regression models, familiarity with proof based mathematics and working knowledge of R. The course starts from 23 Jan 2017.

Prerequisite: Linear Algebra, calculus, statistics and knowledge of R

6.

Advanced Linear Models for Data Science 2: Statistical Linear Models

Duration: 6 weeks

Led by: Johns Hopkins University

This is the second part of the course on advanced linear statistical learning models. For all those who have an understanding of regressions models and are looking to explore this topic further must take this course.

In this course, you will learn about basics of statistical modeling multivariate normal distribution, distributional results, and residuals.

Before you proceed further let me clear, to take this course you need basic understanding of linear algebra & multivariate calculus, statistics & regression models, familiarity with proof based mathematics and working knowledge of R. The course starts from 23 Jan 2017.

Prerequisite: Linear Algebra, calculus, statistics and knowledge of R

7.

Duration: 8 weeks

Led by: University of Notre Dam (edX)

I am someone who is very curious to know how mathematics can be used to drive deeper insights in sports and everyday life.

I came across this course, which shows how your favorite sport uses mathematics to analyze data and know the trends, performance of players and their fellow teams.

In this course, you will learn how inductive reasoning is used in mathematical analysis, how probability is used to evaluate data, assess the risk and outcomes of any event.

All the major team sports, athletic sports, and even extreme sports like mountain climbing have been covered in the course. The course is led by professors of the University of Notre Dam and is currently open for enrolment.

Prerequisite: Statistics & Linear Algebra

## Advanced Mathematics & Statistics

Bravo, by now – you would be on your own. You would have developed a knack for mathematics & statistics and would feel confident about continuous learning – way to go!

1.

Duration: 8 weeks

Led by: University of Melbourne (Coursera)

Every industry & company makes use of optimization. Airlines use optimization to ensure fixed turn-around-time, E-commerce like Amazon uses optimization for on time delivery of products. Macro-level applications of optimization includes deploying electricity to millions of people, way for new medical drug discoveries and many more.

This course provides you a complete understanding of discrete optimization and it is being used in our everyday lives. First, it will take you through fundamental basics of discrete optimization and its various techniques. You will learn about constraint, linear and mixed integer programming. The last section of the course includes advanced topics on optimization.

The prerequisites to take this course are good programming skills, knowledge of fundamental algorithms, and linear algebra. The course starts from 16 Jan 2017 and is conducted by professors at Melbourne University.

Prerequisite: Programming, algorithms and linear algebra

2.

Statistics for Genomic Data Science

Duration: 4 weeks

Led by: Johns Hopkins University

If you aspire to become a generation sequencing data scientist then you must take this course.

In this course, you will learn about exploratory analysis, linear modeling, hypothesis testing & multi-hypothesis testing, different types of process like RNA-seq, GWAS, ChIP-Seq, and DNA Methylation studies. This course is part of Genomic Data Scientist specialization from Johns Hopkins. The course starts from 16 Jan 2017.

This course is part of Genomic Data Scientist specialization from Johns Hopkins. The course starts from 16 Jan 2017.

Prerequisite: Advanced Statistics and algorithms

3.

Biostatistics for Big Data Applications

Duration: 8 weeks

Led by: utmb Health (edX)

This course is an introduction to data analysis using biomedical big data.

In this course, you will learn about fundamental components of biostatistical methods. Working with biomedical big data can pose various challenges for someone not familiar with statistics.

Learn how basic statistics is used in biomedical data types. You will learn about basics of R programming, how to create & interpret graphical summaries of data and inferential statistics for parametric & non-parametric methods. It will provide you hands on experience in R with biomedical problem types.

The course is open for enrolment.

Prerequisite: Advanced statistics and knowledge R

## End Notes

I hope you found this article useful. By now, you would have identified the learning areas for yourself. If you are from mathematics background, you can choose the right courses for yourself. On the other hand, if you do not have a mathematics background, then start from the beginners sections and move ahead.

For those of you, who have taken any of these courses, let us know your feedback about them. Share your opinions with me and other users through comments below. Through this article I wanted to provide you a list of resources available at your disposal in mathematics for data science. Hope you make good use of them.

What is the difference between an event and a random variable? – Quora

**original source : https://www.quora.com/What-is-the-difference-between-an-event-and-a-random-variable**

I assume you’re looking for an informal, intuition-forming, answer rather than the formalism of the mathematical definitions, so here goes:

An event is an outcome or a union of outcomes, when the outcomes are the occurrences over which you can assign probabilities (or measures). A random variable is a variable whose domain is the set of basic events, and whose range (outcome) could be numerical or categorical.

Even though a random variable is a mapping between elementary events and numerical or categorical outcomes, we often analyse random variables by the probabilities that are associated with each value in their range. In other words, we often tend to aggregate the events that yield the same value for the random variable, and we analyse the variable by examining the values that it can take and their corresponding probabilities.

Let’s assume you’re rolling two dice, both being fair dice, and with the rolls being independent of one another.

Events are of the type:

– Got two 6’s (for which the probability would be 1/36).

– Got two odd numbers, i.e., (1, 1) or (3, 3) or (5, 5) or (1, 3) or (1, 5) or (3, 1) or (3, 5) or (5, 1) or (5, 3) (for which the probability would be 9/36 = ¼).

Random variables are of the type:

– The sum of what I got on the two dice.

If I get (1, 1) as the event or outcome, then the variable takes value 2.

If I get (3, 4) as the event or outcome, then the variable takes value 7.

By aggregating over events that yield the same value to the variable, I would analyse it as follows: this random variable takes values between 2 and 12 (its range), each with its corresponding probability:

Value 2 has probability 1/36 because it can only happen if the dice land (1, 1).

Value 12 also has probability 1/36 because it only happens if (6, 6) happens.

Value 7, however, has probability 1/6 because it happens if any of the following dice outcomes is realized: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1).

When you lay out the random variable by the possible values in its range with their corresponding probabilities, you can provide summary statistics such as the expected value of the variable.

For instance, for the two-dice situation, we have:

Value Prob Elementary Events

2 1/36 (1, 1)

3 2/36 (1, 2) (2, 1)

4 3/36 (1, 3) (2, 2) (3, 1)

5 4/36 (1, 4) (2, 3) (3, 2) (4, 1)

6 5/36 (1, 5) (2, 4) (3, 3) (4, 2) (5, 1)

7 6/36 (1, 6) (2, 5) (3, 4) (4, 3) (5, 2) (6, 1)

8 5/36 (2, 6) (3, 5) (4, 4) (5, 3) (6, 2)

9 4/36 (3, 6) (4, 5) (5, 4) (6, 3)

10 3/36 (4, 6) (5, 5) (6, 4)

11 2/36 (5, 6) (6, 5)

12 1/36 (6, 6)

The expected value of my random variable is the weighted sum of the values in its range, with values weighted by their corresponding probabilities, so it is:

(2 * 1/36 + 3 * 2/36 + … + 12 * 1/36) which gives us 7 as the result.

For the sake of completeness, there’s a case where random variable and events collapse into sort of the same thing: it’s when we consider the indicator function of an event as a random variable. So this function takes value 1 if the event in question happens, and it takes value 0 otherwise. Then the expected value of this random variable is simply the probability of that event happening! This is because its expected value is a weighted sum of 1 and 0, with 1 weighted by the probability of the event occurring and 0 weighted by the complementary probability of the event not occurring.

**original source : https://snoopy0505.tistory.com/34**

Proposition (명제)

참과 거짓을 판별할 수 있는 문장.

Axiom (공리)

증명이 필요없는 항상 옳다고 인정되는 명제.

Theorem (정리)

수학적으로 참인 공리 또는 정의를 기반으로 증명된 명제.

**Lemma (보조정리)**

다른 정리를 증명하는 데 쓸 목적으로 증명된 명제.

Corollary (따름정리)

추론이라고도 부른다. 이미 증명된 다른 정리에 의해 바로 유도되는 명제.

출처:

https://snoopy0505.tistory.com/34

[주책이의 작은 공간]

**original source : **

첫번째 성공이 있기까지 실패의 횟수를 x라고 할 때, 또는 첫번째 실패가 있기까지의 성공의 횟수를 x라고할 때 확률변수 x는 기하분포(Geometric distribution)를 따르게 됩니다.

예를들어, 성공확률이 40%인 베르누이시행에서 첫번째 성공이 있기 까지 2번 실패할 확률을 Ms-Excel로 구해보면 다음과 같습니다. Ms-Excel에서는 =negbinomdist function을 사용합니다. 참고로, 베르누이 시행(Bernoulli’s trial)은 결과가 yes, no와 같이 2개의 결과만 있는 서로 독립인 시행을 의미합니다.

=negbinomdist(2,1,40%) = 14.4%

이해를 위해 직접 계산해 볼텐데, 첫번째 성공이 있기까지 두 번 실패한다는 것은, 성공을 1로 표현하고 실패를 0으로 표현하면 다음과 같은 경우입니다.

(0,0,1)

각각의 시행은 독립이므로 결합확률을 구하기 위해서는 각각의 확률을 곱하면 됩니다. 즉, 다음과 같습니다.

60%*60%*40% = 14.4%

성공확률을 p라고 표현하고 실패할 확률을 q(=1-p)라고 표현해 보면 다음과 같습니다..

q*q*p = q^2*p

이를 일반화시켜 볼텐데 실패의 횟수를 x라고 표현해 보면 다음과 같습니다.

q^x * p

이는 기하분포의 공식인데 여기서 보듯이 기하분포의 공식에 시행의 횟수, n은 없습니다. 즉, 기하분포는 시행의 횟수와 관계가 없으며 이를 기하분포의 memoryless property라고 부릅니다. 즉, 기하분포의 확률은 이전에 몇 번 성공했든 또는 이전에 몇 번 실패했든 관계가 없습니다.

어쨌든, 기하분포도 이항분포와 같이 일반적인 확률계산 방식을 공식화해 놓은 것입니다.

기하분포의 모양을 그려볼텐데 파라미터를 변경하면서 Geometric분포의 PMF(Probability Mass Function)와 CDF(Cumulative Distribution Function)를 그려보면 다음과 같습니다.

expectation에 적용 가능한 변형 법칙과 independent의 경우 법칙이 하나가 추가 된다는 것에 유념한다.

inclusion exclusion formula

#합집합 #합

continuous random variables

expectation property

variance

geometric PMF 의 기댓값구하는 방법 E(X) = 1/P를 증명하는 과정설명

https://youtu.be/MuqLI4otMIQ의 내용을 이해하기 힘들어서 아래 링크를 참고했다.

Proof of expected value of geometric random variable | AP Statistics | Khan Academy