diff --git a/Algorithms/Algorithms.pdf b/Algorithms/Algorithms.pdf new file mode 100644 index 0000000..d46d17e Binary files /dev/null and b/Algorithms/Algorithms.pdf differ diff --git a/Artificial Intelligence/Cheat Sheets for AI.pdf b/Artificial Intelligence/Cheat Sheets for AI.pdf new file mode 100644 index 0000000..197065b Binary files /dev/null and b/Artificial Intelligence/Cheat Sheets for AI.pdf differ diff --git a/Artificial Intelligence/README.md b/Artificial Intelligence/README.md new file mode 100644 index 0000000..fd0ee34 --- /dev/null +++ b/Artificial Intelligence/README.md @@ -0,0 +1 @@ +# Artificial Intelligence Cheat Sheet diff --git a/Artificial Intelligence/ai4all.pdf b/Artificial Intelligence/ai4all.pdf new file mode 100644 index 0000000..6268fdf Binary files /dev/null and b/Artificial Intelligence/ai4all.pdf differ diff --git a/Artificial Intelligence/super-cheatsheet-artificial-intelligence.pdf b/Artificial Intelligence/super-cheatsheet-artificial-intelligence.pdf new file mode 100644 index 0000000..fa0348c Binary files /dev/null and b/Artificial Intelligence/super-cheatsheet-artificial-intelligence.pdf differ diff --git a/Big Data/AP.pdf b/Big Data/AP.pdf new file mode 100644 index 0000000..6438925 Binary files /dev/null and b/Big Data/AP.pdf differ diff --git a/Big Data/BIG Data Analytics.pdf b/Big Data/BIG Data Analytics.pdf new file mode 100644 index 0000000..d777f21 Binary files /dev/null and b/Big Data/BIG Data Analytics.pdf differ diff --git a/Big Data/Big-Data-Cheat-Sheet.pdf b/Big Data/Big-Data-Cheat-Sheet.pdf new file mode 100644 index 0000000..82d3862 Binary files /dev/null and b/Big Data/Big-Data-Cheat-Sheet.pdf differ diff --git a/Big Data/Hadoop-and-mapreduce-cheat-sheet.pdf b/Big Data/Hadoop-and-mapreduce-cheat-sheet.pdf new file mode 100644 index 0000000..287de33 Binary files /dev/null and b/Big Data/Hadoop-and-mapreduce-cheat-sheet.pdf differ diff --git a/Big Data/Hortonworks.CheatSheet.SQLtoHive.pdf b/Big 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100644 index 0000000..d7085ab --- /dev/null +++ b/Data Mining/README.md @@ -0,0 +1 @@ +# Data Mining Cheat Sheet diff --git a/Data Mining/What Is Data Warehousing.pdf b/Data Mining/What Is Data Warehousing.pdf new file mode 100644 index 0000000..584664e Binary files /dev/null and b/Data Mining/What Is Data Warehousing.pdf differ diff --git a/Data Mining/What is Dimensional Model.pdf b/Data Mining/What is Dimensional Model.pdf new file mode 100644 index 0000000..8840c9e Binary files /dev/null and b/Data Mining/What is Dimensional Model.pdf differ diff --git a/Data Mining/What is OLAP.pdf b/Data Mining/What is OLAP.pdf new file mode 100644 index 0000000..ab2b647 Binary files /dev/null and b/Data Mining/What is OLAP.pdf differ diff --git a/Data Science/50YearsDataScience-1533532686598.pdf b/Data Science/50YearsDataScience-1533532686598.pdf new file mode 100644 index 0000000..6dc66eb Binary files /dev/null and b/Data Science/50YearsDataScience-1533532686598.pdf differ diff --git a/Data 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index 0000000..f40919a Binary files /dev/null and b/Data Science/Ultimate Guide to Data Cleaning.pdf differ diff --git a/Data Science/VIP Cheat Sheet (ML , DL , AI).pdf b/Data Science/VIP Cheat Sheet (ML , DL , AI).pdf new file mode 100644 index 0000000..d11ba55 Binary files /dev/null and b/Data Science/VIP Cheat Sheet (ML , DL , AI).pdf differ diff --git a/Data Science/What is Exploratory Data Analysis_ - Towards Data Science.pdf b/Data Science/What is Exploratory Data Analysis_ - Towards Data Science.pdf new file mode 100644 index 0000000..120ee76 Binary files /dev/null and b/Data Science/What is Exploratory Data Analysis_ - Towards Data Science.pdf differ diff --git a/Data Science/beginners_guide_to_data_science-_a_twics_guide__1_.pdf b/Data Science/beginners_guide_to_data_science-_a_twics_guide__1_.pdf new file mode 100644 index 0000000..40c78d9 Binary files /dev/null and b/Data Science/beginners_guide_to_data_science-_a_twics_guide__1_.pdf differ diff --git a/Data 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for every dataset.pdf differ diff --git a/Data Visualization/Data Visualization in R.pdf b/Data Visualization/Data Visualization in R.pdf new file mode 100644 index 0000000..f64505b Binary files /dev/null and b/Data Visualization/Data Visualization in R.pdf differ diff --git "a/Data Visualization/Data Visualization \342\200\223 How to Pick the Right Chart Type.pdf" "b/Data Visualization/Data Visualization \342\200\223 How to Pick the Right Chart Type.pdf" new file mode 100644 index 0000000..888cfee Binary files /dev/null and "b/Data Visualization/Data Visualization \342\200\223 How to Pick the Right Chart Type.pdf" differ diff --git a/Data Visualization/Data_Visualization_101_How_to_Design_Charts_and_Graphs.pdf b/Data Visualization/Data_Visualization_101_How_to_Design_Charts_and_Graphs.pdf new file mode 100644 index 0000000..1e76230 Binary files /dev/null and b/Data Visualization/Data_Visualization_101_How_to_Design_Charts_and_Graphs.pdf differ diff --git a/Data 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differ diff --git a/Data Visualization/README.md b/Data Visualization/README.md new file mode 100644 index 0000000..e611918 --- /dev/null +++ b/Data Visualization/README.md @@ -0,0 +1 @@ +# Data Visualization Cheat Sheet diff --git a/Data Visualization/The Art of Effective Visualization of Multi-dimensional Data.pdf b/Data Visualization/The Art of Effective Visualization of Multi-dimensional Data.pdf new file mode 100644 index 0000000..dcfc296 Binary files /dev/null and b/Data Visualization/The Art of Effective Visualization of Multi-dimensional Data.pdf differ diff --git a/Data Visualization/The Next Level of Data Visualization in Python - Towards Data Science.pdf b/Data Visualization/The Next Level of Data Visualization in Python - Towards Data Science.pdf new file mode 100644 index 0000000..5caed0d Binary files /dev/null and b/Data Visualization/The Next Level of Data Visualization in Python - Towards Data Science.pdf differ diff --git a/Data Visualization/Which chart or graph to choose (Tableau).pdf b/Data Visualization/Which chart or graph to choose (Tableau).pdf new file mode 100644 index 0000000..c8cca67 Binary files /dev/null and b/Data Visualization/Which chart or graph to choose (Tableau).pdf differ diff --git a/Data Visualization/data-visualization-storytelling.pdf b/Data Visualization/data-visualization-storytelling.pdf new file mode 100644 index 0000000..6ec0a81 Binary files /dev/null and b/Data Visualization/data-visualization-storytelling.pdf differ diff --git a/Data Visualization/desktop.ini b/Data Visualization/desktop.ini new file mode 100644 index 0000000..bf21110 --- /dev/null +++ b/Data Visualization/desktop.ini @@ -0,0 +1,3 @@ +[LocalizedFileNames] +Awesome Visualization Techniques for every dataset.pdf=@Awesome Visualization Techniques for every dataset.pdf,0 +ML_AndrewNG.pdf=@ML_AndrewNG.pdf,0 diff --git a/Data Visualization/ggplot2.pdf b/Data Visualization/ggplot2.pdf new file mode 100644 index 0000000..ff4b4bb Binary files /dev/null and b/Data Visualization/ggplot2.pdf differ diff --git a/Data Visualization/ggplot2_cheatsheet.pdf b/Data Visualization/ggplot2_cheatsheet.pdf new file mode 100644 index 0000000..7806bfd Binary files /dev/null and b/Data Visualization/ggplot2_cheatsheet.pdf differ diff --git a/Data Visualization/plotly.pdf b/Data Visualization/plotly.pdf new file mode 100644 index 0000000..d9f5fca Binary files /dev/null and b/Data Visualization/plotly.pdf differ diff --git "a/Data Visualization/\342\234\250 Introducing Plotly Express \342\234\250 - Plotly - Medium.pdf" "b/Data Visualization/\342\234\250 Introducing Plotly Express \342\234\250 - Plotly - Medium.pdf" new file mode 100644 index 0000000..7bf70f7 Binary files /dev/null and "b/Data Visualization/\342\234\250 Introducing Plotly Express \342\234\250 - Plotly - Medium.pdf" differ diff --git a/Data Warehouse/Data Warehouse Concepts.pdf b/Data Warehouse/Data Warehouse Concepts.pdf new file mode 100644 index 0000000..f4a34e7 Binary files /dev/null and b/Data Warehouse/Data Warehouse Concepts.pdf differ diff --git a/Data Warehouse/README.md b/Data Warehouse/README.md new file mode 100644 index 0000000..55f63f6 --- /dev/null +++ b/Data Warehouse/README.md @@ -0,0 +1 @@ +# Data Warehouse Cheat Sheet diff --git a/Deep Learning/A Selective Overview of Deep Learning.pdf b/Deep Learning/A Selective Overview of Deep Learning.pdf new file mode 100644 index 0000000..18a5c34 Binary files /dev/null and b/Deep Learning/A Selective Overview of Deep Learning.pdf differ diff --git a/Deep Learning/Coursera Deep Learning course Notes.pdf b/Deep Learning/Coursera Deep Learning course Notes.pdf new file mode 100644 index 0000000..b03cabd Binary files /dev/null and b/Deep Learning/Coursera Deep Learning course Notes.pdf differ diff --git a/Deep Learning/Interview Questions.pdf b/Deep Learning/Interview Questions.pdf new file mode 100644 index 0000000..0527332 Binary files /dev/null and b/Deep Learning/Interview Questions.pdf differ diff --git a/Deep 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0000000..814d083 Binary files /dev/null and b/DevOps/Ansible-Cheat_Sheet_Edureka.png differ diff --git a/DevOps/DevOps.pdf b/DevOps/DevOps.pdf new file mode 100644 index 0000000..abd87ee Binary files /dev/null and b/DevOps/DevOps.pdf differ diff --git a/DevOps/Docker Cheat Sheet.pdf b/DevOps/Docker Cheat Sheet.pdf new file mode 100644 index 0000000..0816a61 Binary files /dev/null and b/DevOps/Docker Cheat Sheet.pdf differ diff --git a/DevOps/README.md b/DevOps/README.md new file mode 100644 index 0000000..8478cc4 --- /dev/null +++ b/DevOps/README.md @@ -0,0 +1 @@ +# DevOps Cheat Sheet diff --git a/DevOps/docker CLI & Dockerfile Cheat Sheet.pdf b/DevOps/docker CLI & Dockerfile Cheat Sheet.pdf new file mode 100644 index 0000000..6795487 Binary files /dev/null and b/DevOps/docker CLI & Dockerfile Cheat Sheet.pdf differ diff --git a/DevOps/docker_cheat sheet.pdf b/DevOps/docker_cheat sheet.pdf new file mode 100644 index 0000000..6795487 Binary files /dev/null and b/DevOps/docker_cheat 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100644 index 0000000..b52b00b Binary files /dev/null and b/Docker & Kubernetes/docker-cheat-sheet.png differ diff --git a/Docker & Kubernetes/kubectl_cheat_sheet.jpg b/Docker & Kubernetes/kubectl_cheat_sheet.jpg new file mode 100644 index 0000000..3abcf0f Binary files /dev/null and b/Docker & Kubernetes/kubectl_cheat_sheet.jpg differ diff --git a/Excel/100 Excel Tips.pdf b/Excel/100 Excel Tips.pdf new file mode 100644 index 0000000..26c8b8f Binary files /dev/null and b/Excel/100 Excel Tips.pdf differ diff --git a/Excel/DAX.pdf b/Excel/DAX.pdf new file mode 100644 index 0000000..0ee7ead Binary files /dev/null and b/Excel/DAX.pdf differ diff --git a/Excel/Excel Cheat Sheet.pdf b/Excel/Excel Cheat Sheet.pdf new file mode 100644 index 0000000..e77172a Binary files /dev/null and b/Excel/Excel Cheat Sheet.pdf differ diff --git a/Excel/Excel Formulas Cheat Sheet.pdf b/Excel/Excel Formulas Cheat Sheet.pdf new file mode 100644 index 0000000..6f0b70b Binary files /dev/null and b/Excel/Excel Formulas Cheat Sheet.pdf differ diff --git a/Excel/Pivot Table.pdf b/Excel/Pivot Table.pdf new file mode 100644 index 0000000..5133961 Binary files /dev/null and b/Excel/Pivot Table.pdf differ diff --git a/Excel/README.md b/Excel/README.md new file mode 100644 index 0000000..f2af373 --- /dev/null +++ b/Excel/README.md @@ -0,0 +1 @@ +# Excel Cheat Sheet diff --git a/Excel/powerpivot.pdf b/Excel/powerpivot.pdf new file mode 100644 index 0000000..097cb12 Binary files /dev/null and b/Excel/powerpivot.pdf differ diff --git a/Git/GitNotesForProfessionals.pdf b/Git/GitNotesForProfessionals.pdf new file mode 100644 index 0000000..5b64084 Binary files /dev/null and b/Git/GitNotesForProfessionals.pdf differ diff --git a/Git/README.md b/Git/README.md new file mode 100644 index 0000000..cf8178a --- /dev/null +++ b/Git/README.md @@ -0,0 +1,2 @@ +# Git Cheat Sheet + diff --git a/Git/atlassian-git-cheatsheet.pdf b/Git/atlassian-git-cheatsheet.pdf new file mode 100644 index 0000000..4f629b2 Binary 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100644 index 0000000..4af7e34 Binary files /dev/null and b/Mathematics/vmls-slides.pdf differ diff --git a/Matlab/README.md b/Matlab/README.md new file mode 100644 index 0000000..6ef3e6e --- /dev/null +++ b/Matlab/README.md @@ -0,0 +1 @@ +# Matlab Cheat Sheet diff --git a/Matlab/matlab_guide.pdf b/Matlab/matlab_guide.pdf new file mode 100644 index 0000000..3578e2a Binary files /dev/null and b/Matlab/matlab_guide.pdf differ diff --git "a/NLP/A Practitioner's Guide to Natural Language Processing (Part I) \342\200\224 Processing & Understanding Text.pdf" "b/NLP/A Practitioner's Guide to Natural Language Processing (Part I) \342\200\224 Processing & Understanding Text.pdf" new file mode 100644 index 0000000..8b4f3d7 Binary files /dev/null and "b/NLP/A Practitioner's Guide to Natural Language Processing (Part I) \342\200\224 Processing & Understanding Text.pdf" differ diff --git a/NLP/Gentle Start to Natural Language Processing using Python.pdf b/NLP/Gentle Start to Natural Language 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Cheat Sheet diff --git a/NLP/desktop.ini b/NLP/desktop.ini new file mode 100644 index 0000000..e09b281 --- /dev/null +++ b/NLP/desktop.ini @@ -0,0 +1,6 @@ +[LocalizedFileNames] +Natural Language Processing is Fun! - Adam Geitgey - Medium.pdf=@Natural Language Processing is Fun! - Adam Geitgey - Medium.pdf,0 +Machine Learning, NLP_ Text Classification using scikit-learn, python and NLTK_.pdf=@Machine Learning, NLP_ Text Classification using scikit-learn, python and NLTK_.pdf,0 +How to solve 90% of NLP problems_ a step-by-step guide.pdf=@How to solve 90% of NLP problems_ a step-by-step guide.pdf,0 +A Practitioner's Guide to Natural Language Processing (Part I) — Processing & Understanding Text.pdf=@A Practitioner's Guide to Natural Language Processing (Part I) — Processing & Understanding Text.pdf,0 +Gentle Start to Natural Language Processing using Python.pdf=@Gentle Start to Natural Language Processing using Python.pdf,0 diff --git a/Numpy/100_numpy_exercises.pdf b/Numpy/100_numpy_exercises.pdf new file mode 100644 index 0000000..ff95006 Binary files /dev/null and b/Numpy/100_numpy_exercises.pdf differ diff --git a/Numpy/README.md b/Numpy/README.md new file mode 100644 index 0000000..5a1dcba --- /dev/null +++ b/Numpy/README.md @@ -0,0 +1 @@ +# Numpy Cheat Sheet diff --git a/Numpy/desktop.ini b/Numpy/desktop.ini new file mode 100644 index 0000000..d0757a2 --- /dev/null +++ b/Numpy/desktop.ini @@ -0,0 +1,4 @@ +[LocalizedFileNames] +100_numpy_exercises.pdf=@100_numpy_exercises.pdf,0 +NumPy Tutorial - A Complete Step-by-Step Guide with clear Examples.pdf=@NumPy Tutorial - A Complete Step-by-Step Guide with clear Examples.pdf,0 +NumPy Tutorial Part 2 - Vital Functions for Data Analysis.pdf=@NumPy Tutorial Part 2 - Vital Functions for Data Analysis.pdf,0 diff --git a/Numpy/numpy_cheat_sheet.pdf b/Numpy/numpy_cheat_sheet.pdf new file mode 100644 index 0000000..daece7c Binary files /dev/null and b/Numpy/numpy_cheat_sheet.pdf differ diff --git a/Ordinary Differential Equations/Linear Algebra and Calculus.pdf b/Ordinary Differential Equations/Linear Algebra and Calculus.pdf new file mode 100644 index 0000000..a7091d8 Binary files /dev/null and b/Ordinary Differential Equations/Linear Algebra and Calculus.pdf differ diff --git a/Ordinary Differential Equations/README.md b/Ordinary Differential Equations/README.md new file mode 100644 index 0000000..2b98f00 --- /dev/null +++ b/Ordinary Differential Equations/README.md @@ -0,0 +1 @@ +# Ordinary Differential Equations Cheat Sheet diff --git a/Ordinary Differential Equations/calculus.pdf b/Ordinary Differential Equations/calculus.pdf new file mode 100644 index 0000000..5cae018 Binary files /dev/null and b/Ordinary Differential Equations/calculus.pdf differ diff --git a/Ordinary Differential Equations/cheatsheet-applications.pdf b/Ordinary Differential Equations/cheatsheet-applications.pdf new file mode 100644 index 0000000..bc52bb7 Binary files /dev/null and b/Ordinary 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+pandas-10min.pdf=@pandas-10min.pdf,0 +Reading and Writing data with PANDAS.pdf=@Reading and Writing data with PANDAS.pdf,0 +pandas_cheat_sheet.pdf=@pandas_cheat_sheet.pdf,0 +Step by Step Guide to Data Cleaning with Python.pdf=@Step by Step Guide to Data Cleaning with Python.pdf,0 diff --git a/Pandas/pandas practice.pdf b/Pandas/pandas practice.pdf new file mode 100644 index 0000000..81365d5 Binary files /dev/null and b/Pandas/pandas practice.pdf differ diff --git a/Pandas/pandas-10min.pdf b/Pandas/pandas-10min.pdf new file mode 100644 index 0000000..79aba6d Binary files /dev/null and b/Pandas/pandas-10min.pdf differ diff --git a/Pandas/pandas_cheat_sheet.pdf b/Pandas/pandas_cheat_sheet.pdf new file mode 100644 index 0000000..806eb0d Binary files /dev/null and b/Pandas/pandas_cheat_sheet.pdf differ diff --git a/Probability/Beginner_Probability_Cheatsheet_Based_On_Khan_Academy_Vids/Probability_Cheatsheet.pdf b/Probability/Beginner_Probability_Cheatsheet_Based_On_Khan_Academy_Vids/Probability_Cheatsheet.pdf new file mode 100644 index 0000000..641fd45 Binary files /dev/null and b/Probability/Beginner_Probability_Cheatsheet_Based_On_Khan_Academy_Vids/Probability_Cheatsheet.pdf differ diff --git a/Probability/Beginner_Probability_Cheatsheet_Based_On_Khan_Academy_Vids/Probability_Cheatsheet.tex b/Probability/Beginner_Probability_Cheatsheet_Based_On_Khan_Academy_Vids/Probability_Cheatsheet.tex new file mode 100644 index 0000000..05b85ca --- /dev/null +++ b/Probability/Beginner_Probability_Cheatsheet_Based_On_Khan_Academy_Vids/Probability_Cheatsheet.tex @@ -0,0 +1,171 @@ +\documentclass[11pt, twocolumn]{article} + +\usepackage[utf8]{inputenc} +\usepackage{graphicx} +\usepackage{amsmath} +\usepackage[a4paper, total={7in, 9in}]{geometry} + + +\title{Probability Cheatsheet} +\author{Yahya~Almardeny\\almardeny@gmail.com} +\date{June 2019} + +\begin{document} +\begin{titlepage} +\maketitle +\end{titlepage} + +\section{Overview} +Probability provides measures for reasoning the likelihood that an event will occur in an experiment. +\section{Terminology} +\begin{enumerate} +\item \textbf{Experiment}: Procedure that yields one of the possible outcomes. E.g. Tossing a coin. +\item \textbf{Sample Space S}: A set of all possible outcomes of an experiment. E.g. S = \{Head, Tail\} when tossing a fair coin. +\item \textbf{Event E}: Set of outcomes of an experiment. Note that if the Event contains only one outcome, it is then called an elementary event. E.g. The event that coin is Head. +\end{enumerate} +\section{Random Variable R.V} +A variable that its possible values are the outcomes of an experiment. It has two types: +\begin{enumerate} +\item \textbf{Discrete R.V}: Take one countable number of distinct values. E.g 1, 2, 3, 4 +\item \textbf{Continuous R.V}: Take infinite number of possible values (usually measurements). E.g. all interval values in the range [1 - 2]. +\end{enumerate} + +\section{Probability of Events} +\begin{enumerate} +\item \textbf{Marginal probability}: Simply $P(A)$ which means "unconditional probability". And for each outcome $s$, it should satisfy: $ 0 \leq P(s) \leq 1$ and $\Sigma P(s_i) = 1$. +\item \textbf{Addition Rule}: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ where $\cup$ means OR, $\cap$ means AND. +\item \textbf{Joint (Compound) Probability Rule}: + \begin{enumerate} + \item Independent Events: $P(A \cap B) = P(A) \times P(B)$. + \item Dependent Events: $P(A \cap B) = P(A) \times P(B | A)$. where $P(B | A)$ read: Probability of B given A had happened. + \end{enumerate} +\item \textbf{Conditional Probability}: $P(A | B) = \frac{P(A \cap B)}{P(B)}$ +\item \textbf{Bayes Theorem}: $P(A | B) = \frac{P(B | A) \times P(A)}{P(B)}$. It gives the probability of A based on prior knowledge of B. +\end{enumerate} + +\section{Probability Distributions} +\begin{enumerate} +\item \textbf{Probability Mass Function (PMF)}: Gives the probability a \textit{discrete} R.V $X$ takes on the value $x$: $P( X = x)$. +\begin{figure}[h!] + \centering + \includegraphics[width=0.7\linewidth]{figs/PMF.png} +\end{figure} +\item \textbf{Probability Density Function (PDF)}: Gives the probability a \textit{continuous} R.V $X$ takes on the value $x$: $P( X = x)$ which means how dense the probability of $X$ near $x$, and it will be the definite Integrals between two points. +\begin{figure}[h!] + \centering + \includegraphics[width=0.6\linewidth]{figs/PDF.png} +\end{figure} +\item \textbf{Cumulative Density Function (CDF)}: Gives the probability a R.V $X$ is less than or equals the value $x$: $P( X \leq x)$. It is cumulative because it adds the probabilities up to $x$. + \begin{enumerate} + \item For Discrete R.V: It is the summation of previous probabilities $\Sigma p(x_i)$. + \item For Continuous R.V: It is the integral $\int_{-\infty}^{x} p(x) dx$ + \end{enumerate} +\end{enumerate} + +\section{Expected Value of R.V} +Intuitively, it is the long-run average value of repetitions of the same experiment it represents (i.e. what outcome to expect on long run) +\begin{enumerate} +\item \textbf{For a Discrete R.V}: It is the average of R.V values based on their associated probabilities $E(X) = \Sigma (x_i \times p(x_i))$. +\item \textbf{For a Continuous R.V}: $E(X) = \int_{-\infty}^{\infty} x f(x) dx$ where $f(x)$ is probability density function. +\end{enumerate} + +\section{Combining Random Variables} +If we have two independent random variables $X$ and $Y$, we can add or subtract to get the total or the difference of their means (expected values) and variances: +$T = X \pm Y \Rightarrow \mu_T = \mu_X \pm \mu_Y ~~ and ~~ \sigma_T^2 = \sigma_X^2 \pm \sigma_Y^2$. + +\section{The Law of Large Numbers} +Let $X$ be R.V of a population where its expected value (i.e Mean) is $E(X)$. Let $\overline{X_n} = \frac{x_1 + x_2 + ... x_n}{n}$ be the mean of $n$ samples $\in X$. As $n \to \infty$ : $\overline{X_n} \to E(X)$. Which means as we do more experiments, the outcomes become closer and closer to the outcomes of the probability theory. + +\section{Statistical Distributions}\footnote{Included because it is related directly to probability} +\begin{enumerate} +\item \textbf{Binomial Distribution}: How many success in finite number of trials: + \begin{itemize} + \item Made up of \textit{independent} trials. + \item Each trial has one of two \textit{discrete} outcomes (success or failure). + \item \textit{Fixed} number of trials + \item Probability of success in each trial is \textit{constant}. + \item E.g. X = number of heads after 10 flips of a coin. + \item \textit{10\% Rule}: We assume independence if the sample $\leq$ 10\% of the population. + \item PMF: $P(X = x) = (_{n}^{k}) p^x (1 - p)^{n-x}$ which means the probability of $k$ successes in $n$ trials. + \item $E(X) = \mu_X = n . p$ where $p$ is the probability of success per trial. + \item $\sigma_X = \sqrt{var(X)} = \sqrt{n . p (1 - p)}$. + \item The more trials we do, the more Binomial becomes Normal distribution. + \end{itemize} + +\item \textbf{Bernoulli Distribution}: Simply a Binomial Distribution with just \textit{one} trial. +\item \textbf{Geometric Distribution}: How many trials until success?. It is similar to Binomial dist. but here we do not have fixed number of trials because we do not know ahead how many trials until we get the desired outcome: + \begin{itemize} + \item PMF: $P(X = x) = (1 - p)^{k - 1} p$. + \item $E(X) = \mu_X = \frac{1}{p}$. + \item $\sigma_X = \sqrt{var(X)} = \sqrt{\frac{1 - p}{p^2}}$. + \end{itemize} + +\item \textbf{Hypergeometric Distribution}: It is a \textit{discrete} probability distribution that describes the probability of $k$ successes in $n$ draws, \textit{without} replacement, from a finite population of size $N$ that contains exactly $K$ objects. In contrast, the binomial distribution describes the probability of $k$ successes in $n$ draws \textit{with} replacement. + +\item \textbf{Poisson Distribution}: It is a \textit{discrete} probability distribution that predicts \textit{rare} events that are independent of one another and occur with a known constant rate $\lambda$. It is usually used instead of Binomial dist. if the number of trials is very high (fixed interval of time) and the probability (occurrence) of each is relatively low: + \begin{itemize} + \item PMF: $P\left( x \right) = \frac{{\lambda ^x }}{{x!}} e^{ - \lambda } $. + \item Mean is $E(X) = \lambda$. and Variance is $\lambda$. + \end{itemize} + +\item \textbf{Normal Distribution (a.k.a Gaussian)}: It is a \textit{continuous} probability distribution that has the notation $X \sim \mathcal{N}(\mu,\,\sigma^{2})$. It is symmetric and has a bell shape where most of the values are near to the mean and follows the 68–95–99.7 rule (a.k.a empirical rule). This rule states that 68.27\%, 95.45\% and 99.73\% of the values lie within one, two and three standard deviations of the mean, respectively: + \begin{itemize} + \item PDF: $P(x) = \frac{1}{{\sigma \sqrt {2\pi } }}e^{{{ - \left( {x - \mu } \right)^2 } \mathord{\left/ {\vphantom {{ - \left( {x - \mu } \right)^2 } {2\sigma ^2 }}} \right. \kern-\nulldelimiterspace} {2\sigma ^2 }}}$. + \item Mean is $E(X) = \mu_X$. and Variance is $\sigma_X^2$. + \end{itemize} + +\item \textbf{Log-Normal Distribution}: It is a \textit{continuous} probability distribution of a random variable whose \textit{logarithm is normally distributed}.Because the values in a log-normal distribution are positive, they create a right-skewed curve which is important in determining which distribution is appropriate to use in investment decision-making. One of the most common applications of log-normal distributions is in the analysis of stock prices. A log-normal distribution can be translated to a normal distribution and vice versa using associated logarithmic calculations. + +\item \textbf{Uniform Distribution}: It is a probability distribution that has \textit{constant} probability. + \begin{enumerate} + \item \textit{Discrete Uniform Dist.}: It takes one of a finite $n$ possible values: + \begin{itemize} + \item PMF: $P(X = x) = \frac{1}{n}$ + \item $E(X) = \mu_X = \frac{(a + b)}{2}$ + \item $\sigma_X^2 =\frac{(b - a + 1)^2 ~ - 1 }{12}$ + \end{itemize} + + \item \textit{Continuous Uniform Dist.}: It takes values within a specified range $a$ to $b$: + \begin{itemize} + \item PDF: + \begin{align*} + P(X = x) = \begin{cases} \frac{1}{b-a}~ if ~ x \in [a , b]\\ 0 ~ otherwise \end{cases} + \end{align*} + \item $E(X) = \mu_X = \frac{1}{2} (a+b)$ + \item $\sigma_X^2 =\frac{1}{12} (b - a)^2 $ + \end{itemize} + + \end{enumerate} + +\item \textbf{Gamma Distribution}: It is a family of a \textit{continuous}, \textit{positive-only}, \textit{unimodal} distributions that encode the time required for $alpha$ events to occur in a Poisson process with mean arrival time of $beta$. These distributions are useful in real-life where something has a natural minimum of $0$. For example, it is commonly used in finance, for elapsed times, or during Poisson processes. In other words, it is a generalization of both the exponential and chi-squared distributions. + +\item \textbf{Exponential Distribution}: Special case of Gamma Distribution. It is a \textit{continuous} analogue of the geometric distribution and often used to model the time elapsed between events. Formally, it is the probability distribution of the time between events in a process where events occur continuously and independently at a \textit{constant} average rate (i.e. Poisson process): + \begin{itemize} + \item PDF: $P(X = x) = {\lambda e}^{-\lambda x}$ + \item $E(X) = \mu_X = \lambda^{-1} = \beta$ + \item $\sigma_X^2 =\lambda^{-2} = \beta^2$ + \end{itemize} + +\item \textbf{Chi-Square Distribution}: It is a special case of the Gamma distribution. It is a \textit{continuous} distribution of a sum of the squares of $k$ independent standard normal deviates (where $k$ is known as degree of freedom). A standard normal deviate is a \textit{random sample} from the standard normal distribution. + \begin{itemize} + \item PDF: $P(X = x) = {\frac{1}{2^{\frac{k}{2}}\Gamma( \frac{K}{2})}}^{x^{\frac{k}{2}-1}~e^{\frac{-x}{2}}}$ where $\Gamma$ is the gamma function + \item $E(X) = \mu_X = k$ + \item $\sigma_X^2 = 2k$ + \end{itemize} + +\item \textbf{Student's t-Distribution}: It is a probability distribution used to estimate population parameters when the sample size is small and standard deviation is unknown. It is mainly used by applying Student's t-test for assessing the statistical significance of the difference between two sample means, and in linear regression analysis. + \begin{itemize} + \item PDF: $$ \frac{\Gamma (\frac{\nu+1}{2})}{\sqrt{\nu \pi} \Gamma (\frac{\nu}{2})} (1 + \frac{x^2}{\nu})^{-\frac{\nu+1}{2}} +$$ where $\nu$ is the number of degrees of freedom and $\Gamma$ is the gamma function. + \item $E(X) = \mu_X = 0 ~ for ~ \nu > 1,$ otherwise undefined. + \item $\sigma_X^2 = \frac{\nu }{\nu -2} ~for~ \nu >2, ~\infty ~for~ 1 < \nu \le 2,$ otherwise undefined. + \end{itemize} +\end{enumerate} + +\end{document} + + + + + + diff --git a/Probability/Beginner_Probability_Cheatsheet_Based_On_Khan_Academy_Vids/figs/PDF.png b/Probability/Beginner_Probability_Cheatsheet_Based_On_Khan_Academy_Vids/figs/PDF.png new file mode 100644 index 0000000..1518688 Binary files /dev/null and b/Probability/Beginner_Probability_Cheatsheet_Based_On_Khan_Academy_Vids/figs/PDF.png differ diff --git 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