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🍒 Hi I'm Cherry! 👋

This is the ultimate deck on Intro to Probability and contains literally everything taught in the free MIT 6.041SC Probabilistic Systems Analysis and Applied Probability course [now renamed 6.3700 Intro to Probability] taught by Prof. John Tsitsiklis.

This course is based on the textbook "Bertsekas, Dimitri, and John Tsitsiklis. Introduction to Probability. 2nd ed. Athena Scientific, 2008. ISBN: 9781886529236"

Download the version of this deck that has images for free on kofi.

📖 Contents of the deck, organized by tags 📖:

contents

⭐️ Features ⭐️:

Cards in the deck contain plentiful derivations, proofs, images, and context on the back to facilitate a deep understanding of concepts and strongly connected memories
Every card is color-coded and math is written in MathJax
Every card includes links to and is tagged by their lecture # in the 6.041SC Probabilistic Systems Analysis and Applied Probability course and RES.6-012 Intro to Probability resource page.
The cards in this deck work with the Clickable Tags addon, which I highly recommend.
All cards are ordered so that material that comes earlier in the course shows up as new cards before material that comes later
Example practice problem cards so you practice and learn the procedure of solving problems (highly effective; will require pen & paper and more time than you may be used to, a few may require calculator)

✏️ Prerequisites for the course and deck 💭:

Calculus
- A strong calculus foundation is necessary, especially optimization which is important in statistical inference
Multivariable Calculus
- Mainly just partial derivatives and double/triple Integrals for analyzing joint distributions of multiple random variables

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To learn how to create amazing cards like I do, check out my 🍒 3 Rules of Card Creation

Sample Data

Text	We're performing a binary hypothesis test andour decision rule has the rejection region {{c3::\(R=\{x\:\|\: L(x)>\xi\} \)::what is \(R\)}}If we {{c1::decrease the critical value \(\xi\)::change \(\xi\) how}}, this:{{c2::\(\uparrow\) probability of rejecting \(H_0\) and accepting \(H_1\)::\(\uparrow\) probability of what decision}}{{c2::\(\uparrow\) false rejection probabiliy \(\alpha(R)\)::change \(\alpha(R)\) how}}{{c2::\(\downarrow\) false acceptance probabiliy \(\beta(R)\)::change \(\beta(R)\) how}}
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Attached	If we increase the critical value \(\xi\), this:\(\uparrow\) probability of accepting \(H_0\) and rejecting \(H_1\)\(\downarrow\) false rejection probabiliy \(\alpha(R)\)\(\uparrow\) false acceptance probabiliy \(\beta(R)\)
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Subject Clozes
Deck ID	74486892
Summary 1	Visual
Back Extra
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Back Extra 2	6.041SC Lecture 24: Classical Inference - IIRES.6-012 Part II: Inference & Limit Theorems
Summary 3	Textbook 9.3: Likelihood Ratio Test
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Summary 4	Textbook 9.3: Neyman-Pearson Lemma
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Summary 5	Textbook 9.3: Binary Hypothesis Testing
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Text	{{c3:: ::deduce the network}}A computer network connects two nodes \(A\) and \(B\) through intermediate nodes \(C, D, E, F,\) as shown above.For every pair of directly connected nodes, say \(i\) and \(j\), there is a given probability \(p_{ij}\) that the link from \(i\) to \(j\) is up (succeeds). We assume {{c2::that link failures/successes are independent of each other::key assumption}}.What is the probability that there is a path connecting \(A\) and \(B\) in which all links are up?{{c1::ask-ans}}
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Answer	We can split the big system into subsystems that are either in series or in parallelNote that\(\mathbf{P}(\text{series subsystem succeeds})=p_1p_2\cdots p_m\)and that\(\begin{align}\mathbf{P}(\text{parallel subsystem succeeds})&=1-\mathbf{P}(\text{parallel subsystem fails})\\&=1-(1-p_1)(1-p_2)\cdots (1-p_m)\end{align}\) So first we find\(\begin{align}\mathbf{P}(C\to B)&= 1-(1-p_{CE}\,p_{EB})(1-p_{CF}\,p_{FB})\\&=1-(1-0.8\cdot 0.9)(1-0.95\cdot 0.85)\\&= 0.946\end{align}\)Then\(\begin{align}\mathbf{P}(A\to B)&=1-(1-p_{AC}\,\mathbf{P}(C\to B))(1-p_{AD}\, p_{DB})\\&= 1 -(1-0.9\cdot 0.946)(1- 0.75\cdot 0.95)\\&= 0.957\end{align}\)
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Subject Clozes
Deck ID	74486892
Summary 1
Back Extra	6.041SC Lecture 3: Independence RES.6-012 Part I: The Fundamentals
Summary 2	Textbook 1.5: Reliability
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Text	Summarize the primary assumptions and goals of {{c1::Bayesian inference::... inference}}.{{c2::ask-ans}}
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Answer	It models the unknown quantity of interest \(\Theta\) as a random variable (or finite collection of random variables) and aims to extract information about \(\Theta\) based on observing a collection \(X=(X_1,\ldots,X_n)\) of related random variables called the observations.It assumes that we know1. the prior distribution \(p_\Theta\) or \(f_\Theta\), depending on whether \(\Theta\) is discrete or continuous2. the conditional distribution \(p_{X\|\Theta}\) or \(f_{X\|\Theta}\), depending on whether \(\Theta\) is discrete or continuousAfter observing \(X=x\), we want to find:\(\to\) the posterior distribution \(p_{\Theta \| X}(\theta \:\|\: X=x)\) by using the appropriate version of Bayes' rule
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Subject Clozes
Deck ID	74486892
Summary 1	Visual
Back Extra
Summary 2	Clarification
Back Extra 2	While we model \(\Theta\) with its prior distribution \(p_\Theta\) or \(f_\Theta\), we don't know the actual value \(\theta\) it's taking, that's why we use our observations \(X=(X_1,\ldots,X_n)\) to narrow down and get a more "accurate" posterior distribution of \(\Theta\).
Summary 3	For comparison: Classical inference
Back Extra 3	Instead of modelling the unknown quantity of interest \(\Theta\) as a random variable (or finite collection of random variables),it models the unknown quantity of interest \(\theta\) as a constant (or finite collection of constants)with the goal of determining what that constant is.
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Back Extra 4	6.041SC Lecture 21: Bayesian Statistical Inference - IRES.6-012 Part II: Inference & Limit Theorems
Summary 5	Textbook 8.1: Bayesian Inference
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Summary 6	Textbook 8.1: Multiparameter Problems
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