\documentclass[11pt]{article}
\usepackage{mathnotes}
\discipline{UCLA-ALIGNED COURSE FIELD NOTES}
\course{STATS 100A}{Introduction to Probability}{Counting, conditioning, random variables, joint laws, expectation, transformations, and limit theorems}

\begin{document}
\makemathcover
\coursefrontmatter{
\begin{curriculumtable}
\unitline{1}{Probability spaces and counting}{Reconstruct the model, work an application, test its assumptions, and explain the result in ordinary language.}
\unitline{2}{Conditional probability, independence, and Bayes}{Reconstruct the model, work an application, test its assumptions, and explain the result in ordinary language.}
\unitline{3}{Discrete random variables}{Reconstruct the model, work an application, test its assumptions, and explain the result in ordinary language.}
\unitline{4}{Continuous random variables}{Reconstruct the model, work an application, test its assumptions, and explain the result in ordinary language.}
\unitline{5}{Joint distributions and random vectors}{Reconstruct the model, work an application, test its assumptions, and explain the result in ordinary language.}
\unitline{6}{Functions and transformations of variables}{Reconstruct the model, work an application, test its assumptions, and explain the result in ordinary language.}
\unitline{7}{Expectation, conditioning, and generating tools}{Reconstruct the model, work an application, test its assumptions, and explain the result in ordinary language.}
\unitline{8}{Laws of large numbers and central limit theorem}{Reconstruct the model, work an application, test its assumptions, and explain the result in ordinary language.}
\unitline{9}{Integration studio}{Combine several chapters in one case, preserve their assumptions, and reconcile the result across representations.}
\unitline{10}{Cumulative review}{Retrieve the full course map from memory and solve mixed problems without chapter labels.}
\end{curriculumtable}
}{\curriculumsource{https://catalog.registrar.ucla.edu/course/2025/STATS100A?year=2025}}

\section{Probability spaces and counting}

\begin{idea}
A probability model specifies outcomes, events, and a measure obeying nonnegativity, normalization, and additivity. Permutations, combinations, inclusion-exclusion, and symmetry calculate finite models only after the sample space is defined correctly.
\end{idea}
% MASTERCLASS EXPANSION
\subsection{Rebuild the chapter from first principles}
The claim I need to own is this: A probability model specifies outcomes, events, and a measure obeying nonnegativity, normalization, and additivity. Permutations, combinations, inclusion-exclusion, and symmetry calculate finite models only after the sample space is defined correctly. I do not count that as learned just because the sentence looks familiar. I should be able to name the objects, state what changes and what stays fixed, recover the rule from the definitions, and explain why the result is allowed.

\begin{move}
\textbf{My working sequence}
\begin{enumerate}
  \item \textbf{Translate.} Rewrite the prompt in the language of probability spaces and counting. List the known quantities, the unknown, the units or types, and any hidden constraint.
  \item \textbf{Choose.} State the theorem, model, algorithm, accounting rule, or physical law before substituting values. This is where I make the assumption visible.
  \item \textbf{Execute.} Work one justified step at a time and keep intermediate state readable enough that I can audit it later.
  \item \textbf{Verify.} Check a boundary case, sign, unit, invariant, order of magnitude, or alternative derivation. Then translate the result back into an ordinary sentence.
\end{enumerate}
\end{move}

\subsection{Details I would refuse to skip}
\begin{itemize}
  \item \textbf{Definitions:} I should be able to define every technical noun in the chapter without using that same noun in the definition.
  \item \textbf{Assumptions:} I write the condition that makes the method legal beside the method itself. A correct procedure under the wrong assumptions is still a wrong answer.
  \item \textbf{Representation:} I move freely among words, equations or code, a diagram, and a small numerical example. If one representation disagrees with another, that disagreement is useful evidence.
  \item \textbf{Failure modes:} I keep the nearest counterexample in mind: the boundary where the rule changes, the invariant breaks, or the model stops matching reality.
  \item \textbf{Communication:} My final answer names what the result means and what it does not establish. I do not let notation do the explanatory work for me.
\end{itemize}

\begin{sanitybox}
\textbf{Closed-note check.} Can I teach probability spaces and counting on a blank page, derive or reconstruct its main rule, work a fresh example, and diagnose a deliberately broken solution? If I can only recognize the finished work, I am not done.
\end{sanitybox}

\subsection{How this chapter connects}
I read this chapter as the bridge from the course foundations to conditional probability, independence, and bayes. The earlier material supplies the language and constraints; this chapter adds a new reasoning move; the later material asks me to use that move without being told its name. That connection matters because exams and real projects mix chapters on purpose.


\subsection{The full working model}
\begin{itemize}
  \item \textbf{Core claim.} A probability model specifies outcomes, events, and a measure obeying nonnegativity, normalization, and additivity.
  \item \textbf{What moves.} Permutations, combinations, inclusion-exclusion, and symmetry calculate finite models only after the sample space is defined correctly.
  \item \textbf{What keeps the answer honest.} The model becomes useful only when I can apply probability spaces and counting to an unfamiliar case and defend the assumptions.
  \item \textbf{Mastery standard.} I can define the objects, rebuild the rule, work a fresh case, compare alternatives, and diagnose a broken solution without relying on recognition.
\end{itemize}

\begin{studenttake}
I write what one equally likely outcome is before counting anything.
\end{studenttake}

\begin{plainbox}
A probability model specifies outcomes, events, and a measure obeying nonnegativity, normalization, and additivity.
\end{plainbox}

\begin{examplebox}
\textbf{Question.} Five cards are drawn from 52 without order. How many hands exist?

\textbf{Walkthrough.} There are 52 choose 5, equal to 2,598,960. Combinations apply because order within a hand does not matter.
\end{examplebox}

\begin{warningbox}
The fastest way to get this chapter wrong is to use the visible procedure without naming its assumption, units, time period, reference group, or boundary. I put that condition beside the work and verify the result independently.
\end{warningbox}

\begin{chapterpractice}
\textbf{Practice 1.} Five cards are drawn from 52 without order. How many hands exist?

\textbf{Solution.} There are 52 choose 5, equal to 2,598,960. Combinations apply because order within a hand does not matter.

\medskip
\textbf{Practice 2.} Explain probability spaces and counting to a classmate using one definition, one concrete example, one assumption, and one failure mode.

\textbf{Solution.} Begin with the core claim above, use the worked case as the concrete example, state the condition highlighted by the warning, and choose a failure where that condition does not hold. A complete explanation connects all four pieces instead of listing them.

\medskip
\textbf{Mastery practice A.} Reconstruct probability spaces and counting from a blank page. Include the central definition, the governing rule, one required assumption, and one boundary case.

\textbf{Solution.} Start from the blue idea box, but rewrite it in my own words. Identify the objects and state, name the rule before using it, copy the relevant assumption from the details audit, and choose a boundary where that assumption becomes equality or fails. A complete solution explains why each part belongs; a list of vocabulary is not enough.

\medskip
\textbf{Mastery practice B.} A classmate gets a polished-looking answer to a probability spaces and counting problem but never states a model or checks the result. Write the shortest useful review of that solution.

\textbf{Solution.} I would ask four concrete questions: What exactly are the inputs and desired output? Which rule licenses the main step? Which assumption could invalidate it? Which independent check supports the conclusion? If the answer cannot survive those four questions, I treat the numerical or verbal result as unverified.

\end{chapterpractice}

\section{Conditional probability, independence, and Bayes}

\begin{idea}
Conditioning restricts the sample space using new information. Independence is the factorization of joint probability, not the absence of any relationship in ordinary language; Bayes' rule reverses conditioning using base rates.
\end{idea}
% MASTERCLASS EXPANSION
\subsection{Rebuild the chapter from first principles}
The claim I need to own is this: Conditioning restricts the sample space using new information. Independence is the factorization of joint probability, not the absence of any relationship in ordinary language; Bayes' rule reverses conditioning using base rates. I do not count that as learned just because the sentence looks familiar. I should be able to name the objects, state what changes and what stays fixed, recover the rule from the definitions, and explain why the result is allowed.

\begin{move}
\textbf{My working sequence}
\begin{enumerate}
  \item \textbf{Translate.} Rewrite the prompt in the language of conditional probability, independence, and bayes. List the known quantities, the unknown, the units or types, and any hidden constraint.
  \item \textbf{Choose.} State the theorem, model, algorithm, accounting rule, or physical law before substituting values. This is where I make the assumption visible.
  \item \textbf{Execute.} Work one justified step at a time and keep intermediate state readable enough that I can audit it later.
  \item \textbf{Verify.} Check a boundary case, sign, unit, invariant, order of magnitude, or alternative derivation. Then translate the result back into an ordinary sentence.
\end{enumerate}
\end{move}

\subsection{Details I would refuse to skip}
\begin{itemize}
  \item \textbf{Definitions:} I should be able to define every technical noun in the chapter without using that same noun in the definition.
  \item \textbf{Assumptions:} I write the condition that makes the method legal beside the method itself. A correct procedure under the wrong assumptions is still a wrong answer.
  \item \textbf{Representation:} I move freely among words, equations or code, a diagram, and a small numerical example. If one representation disagrees with another, that disagreement is useful evidence.
  \item \textbf{Failure modes:} I keep the nearest counterexample in mind: the boundary where the rule changes, the invariant breaks, or the model stops matching reality.
  \item \textbf{Communication:} My final answer names what the result means and what it does not establish. I do not let notation do the explanatory work for me.
\end{itemize}

\begin{sanitybox}
\textbf{Closed-note check.} Can I teach conditional probability, independence, and bayes on a blank page, derive or reconstruct its main rule, work a fresh example, and diagnose a deliberately broken solution? If I can only recognize the finished work, I am not done.
\end{sanitybox}

\subsection{How this chapter connects}
I read this chapter as the bridge from probability spaces and counting to discrete random variables. The earlier material supplies the language and constraints; this chapter adds a new reasoning move; the later material asks me to use that move without being told its name. That connection matters because exams and real projects mix chapters on purpose.


\subsection{The full working model}
\begin{itemize}
  \item \textbf{Core claim.} Conditioning restricts the sample space using new information.
  \item \textbf{What moves.} Independence is the factorization of joint probability, not the absence of any relationship in ordinary language; Bayes' rule reverses conditioning using base rates.
  \item \textbf{What keeps the answer honest.} The model becomes useful only when I can apply conditional probability, independence, and bayes to an unfamiliar case and defend the assumptions.
  \item \textbf{Mastery standard.} I can define the objects, rebuild the rule, work a fresh case, compare alternatives, and diagnose a broken solution without relying on recognition.
\end{itemize}

\begin{studenttake}
I keep the denominator event visible; most conditional mistakes come from silently changing the reference population.
\end{studenttake}

\begin{plainbox}
Conditioning restricts the sample space using new information.
\end{plainbox}

\begin{examplebox}
\textbf{Question.} A disease has 1 percent prevalence, 90 percent sensitivity, and 5 percent false-positive rate. Find positive predictive value.

\textbf{Walkthrough.} Bayes gives 0.009 divided by 0.009+0.0495, approximately 15.4 percent. The low base rate dominates despite good sensitivity.
\end{examplebox}

\begin{warningbox}
The fastest way to get this chapter wrong is to use the visible procedure without naming its assumption, units, time period, reference group, or boundary. I put that condition beside the work and verify the result independently.
\end{warningbox}

\begin{chapterpractice}
\textbf{Practice 1.} A disease has 1 percent prevalence, 90 percent sensitivity, and 5 percent false-positive rate. Find positive predictive value.

\textbf{Solution.} Bayes gives 0.009 divided by 0.009+0.0495, approximately 15.4 percent. The low base rate dominates despite good sensitivity.

\medskip
\textbf{Practice 2.} Explain conditional probability, independence, and bayes to a classmate using one definition, one concrete example, one assumption, and one failure mode.

\textbf{Solution.} Begin with the core claim above, use the worked case as the concrete example, state the condition highlighted by the warning, and choose a failure where that condition does not hold. A complete explanation connects all four pieces instead of listing them.

\medskip
\textbf{Mastery practice A.} Reconstruct conditional probability, independence, and bayes from a blank page. Include the central definition, the governing rule, one required assumption, and one boundary case.

\textbf{Solution.} Start from the blue idea box, but rewrite it in my own words. Identify the objects and state, name the rule before using it, copy the relevant assumption from the details audit, and choose a boundary where that assumption becomes equality or fails. A complete solution explains why each part belongs; a list of vocabulary is not enough.

\medskip
\textbf{Mastery practice B.} A classmate gets a polished-looking answer to a conditional probability, independence, and bayes problem but never states a model or checks the result. Write the shortest useful review of that solution.

\textbf{Solution.} I would ask four concrete questions: What exactly are the inputs and desired output? Which rule licenses the main step? Which assumption could invalidate it? Which independent check supports the conclusion? If the answer cannot survive those four questions, I treat the numerical or verbal result as unverified.

\end{chapterpractice}

\section{Discrete random variables}

\begin{idea}
A discrete random variable maps outcomes to countable values. Probability mass functions, cumulative distributions, expectation, variance, Bernoulli, binomial, geometric, negative binomial, hypergeometric, and Poisson models encode different trial structures.
\end{idea}
% MASTERCLASS EXPANSION
\subsection{Rebuild the chapter from first principles}
The claim I need to own is this: A discrete random variable maps outcomes to countable values. Probability mass functions, cumulative distributions, expectation, variance, Bernoulli, binomial, geometric, negative binomial, hypergeometric, and Poisson models encode different trial structures. I do not count that as learned just because the sentence looks familiar. I should be able to name the objects, state what changes and what stays fixed, recover the rule from the definitions, and explain why the result is allowed.

\begin{move}
\textbf{My working sequence}
\begin{enumerate}
  \item \textbf{Translate.} Rewrite the prompt in the language of discrete random variables. List the known quantities, the unknown, the units or types, and any hidden constraint.
  \item \textbf{Choose.} State the theorem, model, algorithm, accounting rule, or physical law before substituting values. This is where I make the assumption visible.
  \item \textbf{Execute.} Work one justified step at a time and keep intermediate state readable enough that I can audit it later.
  \item \textbf{Verify.} Check a boundary case, sign, unit, invariant, order of magnitude, or alternative derivation. Then translate the result back into an ordinary sentence.
\end{enumerate}
\end{move}

\subsection{Details I would refuse to skip}
\begin{itemize}
  \item \textbf{Definitions:} I should be able to define every technical noun in the chapter without using that same noun in the definition.
  \item \textbf{Assumptions:} I write the condition that makes the method legal beside the method itself. A correct procedure under the wrong assumptions is still a wrong answer.
  \item \textbf{Representation:} I move freely among words, equations or code, a diagram, and a small numerical example. If one representation disagrees with another, that disagreement is useful evidence.
  \item \textbf{Failure modes:} I keep the nearest counterexample in mind: the boundary where the rule changes, the invariant breaks, or the model stops matching reality.
  \item \textbf{Communication:} My final answer names what the result means and what it does not establish. I do not let notation do the explanatory work for me.
\end{itemize}

\begin{sanitybox}
\textbf{Closed-note check.} Can I teach discrete random variables on a blank page, derive or reconstruct its main rule, work a fresh example, and diagnose a deliberately broken solution? If I can only recognize the finished work, I am not done.
\end{sanitybox}

\subsection{How this chapter connects}
I read this chapter as the bridge from conditional probability, independence, and bayes to continuous random variables. The earlier material supplies the language and constraints; this chapter adds a new reasoning move; the later material asks me to use that move without being told its name. That connection matters because exams and real projects mix chapters on purpose.


\subsection{The full working model}
\begin{itemize}
  \item \textbf{Core claim.} A discrete random variable maps outcomes to countable values.
  \item \textbf{What moves.} Probability mass functions, cumulative distributions, expectation, variance, Bernoulli, binomial, geometric, negative binomial, hypergeometric, and Poisson models encode different trial structures.
  \item \textbf{What keeps the answer honest.} The model becomes useful only when I can apply discrete random variables to an unfamiliar case and defend the assumptions.
  \item \textbf{Mastery standard.} I can define the objects, rebuild the rule, work a fresh case, compare alternatives, and diagnose a broken solution without relying on recognition.
\end{itemize}

\begin{studenttake}
I choose a distribution from the data-generating story, not because its formula looks familiar.
\end{studenttake}

\begin{plainbox}
A discrete random variable maps outcomes to countable values.
\end{plainbox}

\begin{examplebox}
\textbf{Question.} For X binomial with n=20 and p=0.3, find mean and variance.

\textbf{Walkthrough.} The mean is np=6 and variance is np(1-p)=4.2. Independence and constant success probability are part of the model.
\end{examplebox}

\begin{warningbox}
The fastest way to get this chapter wrong is to use the visible procedure without naming its assumption, units, time period, reference group, or boundary. I put that condition beside the work and verify the result independently.
\end{warningbox}

\begin{chapterpractice}
\textbf{Practice 1.} For X binomial with n=20 and p=0.3, find mean and variance.

\textbf{Solution.} The mean is np=6 and variance is np(1-p)=4.2. Independence and constant success probability are part of the model.

\medskip
\textbf{Practice 2.} Explain discrete random variables to a classmate using one definition, one concrete example, one assumption, and one failure mode.

\textbf{Solution.} Begin with the core claim above, use the worked case as the concrete example, state the condition highlighted by the warning, and choose a failure where that condition does not hold. A complete explanation connects all four pieces instead of listing them.

\medskip
\textbf{Mastery practice A.} Reconstruct discrete random variables from a blank page. Include the central definition, the governing rule, one required assumption, and one boundary case.

\textbf{Solution.} Start from the blue idea box, but rewrite it in my own words. Identify the objects and state, name the rule before using it, copy the relevant assumption from the details audit, and choose a boundary where that assumption becomes equality or fails. A complete solution explains why each part belongs; a list of vocabulary is not enough.

\medskip
\textbf{Mastery practice B.} A classmate gets a polished-looking answer to a discrete random variables problem but never states a model or checks the result. Write the shortest useful review of that solution.

\textbf{Solution.} I would ask four concrete questions: What exactly are the inputs and desired output? Which rule licenses the main step? Which assumption could invalidate it? Which independent check supports the conclusion? If the answer cannot survive those four questions, I treat the numerical or verbal result as unverified.

\end{chapterpractice}

\section{Continuous random variables}

\begin{idea}
Densities integrate to probability and may exceed one at a point; point probability is zero for continuous laws. Uniform, exponential, gamma, beta, and normal distributions model different supports, shapes, and memory structures.
\end{idea}
% MASTERCLASS EXPANSION
\subsection{Rebuild the chapter from first principles}
The claim I need to own is this: Densities integrate to probability and may exceed one at a point; point probability is zero for continuous laws. Uniform, exponential, gamma, beta, and normal distributions model different supports, shapes, and memory structures. I do not count that as learned just because the sentence looks familiar. I should be able to name the objects, state what changes and what stays fixed, recover the rule from the definitions, and explain why the result is allowed.

\begin{move}
\textbf{My working sequence}
\begin{enumerate}
  \item \textbf{Translate.} Rewrite the prompt in the language of continuous random variables. List the known quantities, the unknown, the units or types, and any hidden constraint.
  \item \textbf{Choose.} State the theorem, model, algorithm, accounting rule, or physical law before substituting values. This is where I make the assumption visible.
  \item \textbf{Execute.} Work one justified step at a time and keep intermediate state readable enough that I can audit it later.
  \item \textbf{Verify.} Check a boundary case, sign, unit, invariant, order of magnitude, or alternative derivation. Then translate the result back into an ordinary sentence.
\end{enumerate}
\end{move}

\subsection{Details I would refuse to skip}
\begin{itemize}
  \item \textbf{Definitions:} I should be able to define every technical noun in the chapter without using that same noun in the definition.
  \item \textbf{Assumptions:} I write the condition that makes the method legal beside the method itself. A correct procedure under the wrong assumptions is still a wrong answer.
  \item \textbf{Representation:} I move freely among words, equations or code, a diagram, and a small numerical example. If one representation disagrees with another, that disagreement is useful evidence.
  \item \textbf{Failure modes:} I keep the nearest counterexample in mind: the boundary where the rule changes, the invariant breaks, or the model stops matching reality.
  \item \textbf{Communication:} My final answer names what the result means and what it does not establish. I do not let notation do the explanatory work for me.
\end{itemize}

\begin{sanitybox}
\textbf{Closed-note check.} Can I teach continuous random variables on a blank page, derive or reconstruct its main rule, work a fresh example, and diagnose a deliberately broken solution? If I can only recognize the finished work, I am not done.
\end{sanitybox}

\subsection{How this chapter connects}
I read this chapter as the bridge from discrete random variables to joint distributions and random vectors. The earlier material supplies the language and constraints; this chapter adds a new reasoning move; the later material asks me to use that move without being told its name. That connection matters because exams and real projects mix chapters on purpose.


\subsection{The full working model}
\begin{itemize}
  \item \textbf{Core claim.} Densities integrate to probability and may exceed one at a point; point probability is zero for continuous laws.
  \item \textbf{What moves.} Uniform, exponential, gamma, beta, and normal distributions model different supports, shapes, and memory structures.
  \item \textbf{What keeps the answer honest.} The model becomes useful only when I can apply continuous random variables to an unfamiliar case and defend the assumptions.
  \item \textbf{Mastery standard.} I can define the objects, rebuild the rule, work a fresh case, compare alternatives, and diagnose a broken solution without relying on recognition.
\end{itemize}

\begin{studenttake}
My shorthand for this chapter is: my shorthand for this chapter is: a density is height, not probability; area is probability.
\end{studenttake}

\begin{plainbox}
Densities integrate to probability and may exceed one at a point; point probability is zero for continuous laws.
\end{plainbox}

\begin{examplebox}
\textbf{Question.} If X is exponential with rate 2, find P(X\textgreater{}1).

\textbf{Walkthrough.} The survival probability is exp(-2 times 1), or about 0.1353. Exponential waiting time is memoryless.
\end{examplebox}

\begin{warningbox}
The fastest way to get this chapter wrong is to use the visible procedure without naming its assumption, units, time period, reference group, or boundary. I put that condition beside the work and verify the result independently.
\end{warningbox}

\begin{chapterpractice}
\textbf{Practice 1.} If X is exponential with rate 2, find P(X\textgreater{}1).

\textbf{Solution.} The survival probability is exp(-2 times 1), or about 0.1353. Exponential waiting time is memoryless.

\medskip
\textbf{Practice 2.} Explain continuous random variables to a classmate using one definition, one concrete example, one assumption, and one failure mode.

\textbf{Solution.} Begin with the core claim above, use the worked case as the concrete example, state the condition highlighted by the warning, and choose a failure where that condition does not hold. A complete explanation connects all four pieces instead of listing them.

\medskip
\textbf{Mastery practice A.} Reconstruct continuous random variables from a blank page. Include the central definition, the governing rule, one required assumption, and one boundary case.

\textbf{Solution.} Start from the blue idea box, but rewrite it in my own words. Identify the objects and state, name the rule before using it, copy the relevant assumption from the details audit, and choose a boundary where that assumption becomes equality or fails. A complete solution explains why each part belongs; a list of vocabulary is not enough.

\medskip
\textbf{Mastery practice B.} A classmate gets a polished-looking answer to a continuous random variables problem but never states a model or checks the result. Write the shortest useful review of that solution.

\textbf{Solution.} I would ask four concrete questions: What exactly are the inputs and desired output? Which rule licenses the main step? Which assumption could invalidate it? Which independent check supports the conclusion? If the answer cannot survive those four questions, I treat the numerical or verbal result as unverified.

\end{chapterpractice}

\section{Joint distributions and random vectors}

\begin{idea}
Joint, marginal, and conditional laws describe several variables. Covariance and correlation measure linear co-movement but not general dependence; independence implies zero covariance when moments exist, not conversely.
\end{idea}
% MASTERCLASS EXPANSION
\subsection{Rebuild the chapter from first principles}
The claim I need to own is this: Joint, marginal, and conditional laws describe several variables. Covariance and correlation measure linear co-movement but not general dependence; independence implies zero covariance when moments exist, not conversely. I do not count that as learned just because the sentence looks familiar. I should be able to name the objects, state what changes and what stays fixed, recover the rule from the definitions, and explain why the result is allowed.

\begin{move}
\textbf{My working sequence}
\begin{enumerate}
  \item \textbf{Translate.} Rewrite the prompt in the language of joint distributions and random vectors. List the known quantities, the unknown, the units or types, and any hidden constraint.
  \item \textbf{Choose.} State the theorem, model, algorithm, accounting rule, or physical law before substituting values. This is where I make the assumption visible.
  \item \textbf{Execute.} Work one justified step at a time and keep intermediate state readable enough that I can audit it later.
  \item \textbf{Verify.} Check a boundary case, sign, unit, invariant, order of magnitude, or alternative derivation. Then translate the result back into an ordinary sentence.
\end{enumerate}
\end{move}

\subsection{Details I would refuse to skip}
\begin{itemize}
  \item \textbf{Definitions:} I should be able to define every technical noun in the chapter without using that same noun in the definition.
  \item \textbf{Assumptions:} I write the condition that makes the method legal beside the method itself. A correct procedure under the wrong assumptions is still a wrong answer.
  \item \textbf{Representation:} I move freely among words, equations or code, a diagram, and a small numerical example. If one representation disagrees with another, that disagreement is useful evidence.
  \item \textbf{Failure modes:} I keep the nearest counterexample in mind: the boundary where the rule changes, the invariant breaks, or the model stops matching reality.
  \item \textbf{Communication:} My final answer names what the result means and what it does not establish. I do not let notation do the explanatory work for me.
\end{itemize}

\begin{sanitybox}
\textbf{Closed-note check.} Can I teach joint distributions and random vectors on a blank page, derive or reconstruct its main rule, work a fresh example, and diagnose a deliberately broken solution? If I can only recognize the finished work, I am not done.
\end{sanitybox}

\subsection{How this chapter connects}
I read this chapter as the bridge from continuous random variables to functions and transformations of variables. The earlier material supplies the language and constraints; this chapter adds a new reasoning move; the later material asks me to use that move without being told its name. That connection matters because exams and real projects mix chapters on purpose.


\subsection{The full working model}
\begin{itemize}
  \item \textbf{Core claim.} Joint, marginal, and conditional laws describe several variables.
  \item \textbf{What moves.} Covariance and correlation measure linear co-movement but not general dependence; independence implies zero covariance when moments exist, not conversely.
  \item \textbf{What keeps the answer honest.} The model becomes useful only when I can apply joint distributions and random vectors to an unfamiliar case and defend the assumptions.
  \item \textbf{Mastery standard.} I can define the objects, rebuild the rule, work a fresh case, compare alternatives, and diagnose a broken solution without relying on recognition.
\end{itemize}

\begin{studenttake}
My shorthand for this chapter is: my shorthand for this chapter is: a correlation of zero is a statement about one pattern, not a guarantee the variables ignore each other.
\end{studenttake}

\begin{plainbox}
Joint, marginal, and conditional laws describe several variables.
\end{plainbox}

\begin{examplebox}
\textbf{Question.} If Cov(X,Y)=3, Var(X)=4, and Var(Y)=9, find correlation.

\textbf{Walkthrough.} Correlation is 3 divided by sqrt(4 times 9), or 0.5.
\end{examplebox}

\begin{warningbox}
The fastest way to get this chapter wrong is to use the visible procedure without naming its assumption, units, time period, reference group, or boundary. I put that condition beside the work and verify the result independently.
\end{warningbox}

\begin{chapterpractice}
\textbf{Practice 1.} If Cov(X,Y)=3, Var(X)=4, and Var(Y)=9, find correlation.

\textbf{Solution.} Correlation is 3 divided by sqrt(4 times 9), or 0.5.

\medskip
\textbf{Practice 2.} Explain joint distributions and random vectors to a classmate using one definition, one concrete example, one assumption, and one failure mode.

\textbf{Solution.} Begin with the core claim above, use the worked case as the concrete example, state the condition highlighted by the warning, and choose a failure where that condition does not hold. A complete explanation connects all four pieces instead of listing them.

\medskip
\textbf{Mastery practice A.} Reconstruct joint distributions and random vectors from a blank page. Include the central definition, the governing rule, one required assumption, and one boundary case.

\textbf{Solution.} Start from the blue idea box, but rewrite it in my own words. Identify the objects and state, name the rule before using it, copy the relevant assumption from the details audit, and choose a boundary where that assumption becomes equality or fails. A complete solution explains why each part belongs; a list of vocabulary is not enough.

\medskip
\textbf{Mastery practice B.} A classmate gets a polished-looking answer to a joint distributions and random vectors problem but never states a model or checks the result. Write the shortest useful review of that solution.

\textbf{Solution.} I would ask four concrete questions: What exactly are the inputs and desired output? Which rule licenses the main step? Which assumption could invalidate it? Which independent check supports the conclusion? If the answer cannot survive those four questions, I treat the numerical or verbal result as unverified.

\end{chapterpractice}

\section{Functions and transformations of variables}

\begin{idea}
Transformations require mapping support and probability correctly. CDF, change-of-variables, Jacobian, convolution, and order-statistic methods derive distributions of functions, sums, ratios, minima, and maxima.
\end{idea}
% MASTERCLASS EXPANSION
\subsection{Rebuild the chapter from first principles}
The claim I need to own is this: Transformations require mapping support and probability correctly. CDF, change-of-variables, Jacobian, convolution, and order-statistic methods derive distributions of functions, sums, ratios, minima, and maxima. I do not count that as learned just because the sentence looks familiar. I should be able to name the objects, state what changes and what stays fixed, recover the rule from the definitions, and explain why the result is allowed.

\begin{move}
\textbf{My working sequence}
\begin{enumerate}
  \item \textbf{Translate.} Rewrite the prompt in the language of functions and transformations of variables. List the known quantities, the unknown, the units or types, and any hidden constraint.
  \item \textbf{Choose.} State the theorem, model, algorithm, accounting rule, or physical law before substituting values. This is where I make the assumption visible.
  \item \textbf{Execute.} Work one justified step at a time and keep intermediate state readable enough that I can audit it later.
  \item \textbf{Verify.} Check a boundary case, sign, unit, invariant, order of magnitude, or alternative derivation. Then translate the result back into an ordinary sentence.
\end{enumerate}
\end{move}

\subsection{Details I would refuse to skip}
\begin{itemize}
  \item \textbf{Definitions:} I should be able to define every technical noun in the chapter without using that same noun in the definition.
  \item \textbf{Assumptions:} I write the condition that makes the method legal beside the method itself. A correct procedure under the wrong assumptions is still a wrong answer.
  \item \textbf{Representation:} I move freely among words, equations or code, a diagram, and a small numerical example. If one representation disagrees with another, that disagreement is useful evidence.
  \item \textbf{Failure modes:} I keep the nearest counterexample in mind: the boundary where the rule changes, the invariant breaks, or the model stops matching reality.
  \item \textbf{Communication:} My final answer names what the result means and what it does not establish. I do not let notation do the explanatory work for me.
\end{itemize}

\begin{sanitybox}
\textbf{Closed-note check.} Can I teach functions and transformations of variables on a blank page, derive or reconstruct its main rule, work a fresh example, and diagnose a deliberately broken solution? If I can only recognize the finished work, I am not done.
\end{sanitybox}

\subsection{How this chapter connects}
I read this chapter as the bridge from joint distributions and random vectors to expectation, conditioning, and generating tools. The earlier material supplies the language and constraints; this chapter adds a new reasoning move; the later material asks me to use that move without being told its name. That connection matters because exams and real projects mix chapters on purpose.


\subsection{The full working model}
\begin{itemize}
  \item \textbf{Core claim.} Transformations require mapping support and probability correctly.
  \item \textbf{What moves.} CDF, change-of-variables, Jacobian, convolution, and order-statistic methods derive distributions of functions, sums, ratios, minima, and maxima.
  \item \textbf{What keeps the answer honest.} The model becomes useful only when I can apply functions and transformations of variables to an unfamiliar case and defend the assumptions.
  \item \textbf{Mastery standard.} I can define the objects, rebuild the rule, work a fresh case, compare alternatives, and diagnose a broken solution without relying on recognition.
\end{itemize}

\begin{studenttake}
I draw the transformed support before touching the Jacobian.
\end{studenttake}

\begin{plainbox}
Transformations require mapping support and probability correctly.
\end{plainbox}

\begin{examplebox}
\textbf{Question.} If X is uniform on 0 to 1 and Y=X squared, find the CDF of Y on 0 to 1.

\textbf{Walkthrough.} P(Y\textless{}=y)=P(X\textless{}=sqrt(y))=sqrt(y). Differentiating gives density 1 over 2sqrt(y) for 0\textless{}y\textless{}1.
\end{examplebox}

\begin{warningbox}
The fastest way to get this chapter wrong is to use the visible procedure without naming its assumption, units, time period, reference group, or boundary. I put that condition beside the work and verify the result independently.
\end{warningbox}

\begin{chapterpractice}
\textbf{Practice 1.} If X is uniform on 0 to 1 and Y=X squared, find the CDF of Y on 0 to 1.

\textbf{Solution.} P(Y\textless{}=y)=P(X\textless{}=sqrt(y))=sqrt(y). Differentiating gives density 1 over 2sqrt(y) for 0\textless{}y\textless{}1.

\medskip
\textbf{Practice 2.} Explain functions and transformations of variables to a classmate using one definition, one concrete example, one assumption, and one failure mode.

\textbf{Solution.} Begin with the core claim above, use the worked case as the concrete example, state the condition highlighted by the warning, and choose a failure where that condition does not hold. A complete explanation connects all four pieces instead of listing them.

\medskip
\textbf{Mastery practice A.} Reconstruct functions and transformations of variables from a blank page. Include the central definition, the governing rule, one required assumption, and one boundary case.

\textbf{Solution.} Start from the blue idea box, but rewrite it in my own words. Identify the objects and state, name the rule before using it, copy the relevant assumption from the details audit, and choose a boundary where that assumption becomes equality or fails. A complete solution explains why each part belongs; a list of vocabulary is not enough.

\medskip
\textbf{Mastery practice B.} A classmate gets a polished-looking answer to a functions and transformations of variables problem but never states a model or checks the result. Write the shortest useful review of that solution.

\textbf{Solution.} I would ask four concrete questions: What exactly are the inputs and desired output? Which rule licenses the main step? Which assumption could invalidate it? Which independent check supports the conclusion? If the answer cannot survive those four questions, I treat the numerical or verbal result as unverified.

\end{chapterpractice}

\section{Expectation, conditioning, and generating tools}

\begin{idea}
Linearity of expectation needs no independence. Conditional expectation is a random variable and projection; total expectation and total variance decompose uncertainty. Moment and probability generating functions encode distributions when they exist.
\end{idea}
% MASTERCLASS EXPANSION
\subsection{Rebuild the chapter from first principles}
The claim I need to own is this: Linearity of expectation needs no independence. Conditional expectation is a random variable and projection; total expectation and total variance decompose uncertainty. Moment and probability generating functions encode distributions when they exist. I do not count that as learned just because the sentence looks familiar. I should be able to name the objects, state what changes and what stays fixed, recover the rule from the definitions, and explain why the result is allowed.

\begin{move}
\textbf{My working sequence}
\begin{enumerate}
  \item \textbf{Translate.} Rewrite the prompt in the language of expectation, conditioning, and generating tools. List the known quantities, the unknown, the units or types, and any hidden constraint.
  \item \textbf{Choose.} State the theorem, model, algorithm, accounting rule, or physical law before substituting values. This is where I make the assumption visible.
  \item \textbf{Execute.} Work one justified step at a time and keep intermediate state readable enough that I can audit it later.
  \item \textbf{Verify.} Check a boundary case, sign, unit, invariant, order of magnitude, or alternative derivation. Then translate the result back into an ordinary sentence.
\end{enumerate}
\end{move}

\subsection{Details I would refuse to skip}
\begin{itemize}
  \item \textbf{Definitions:} I should be able to define every technical noun in the chapter without using that same noun in the definition.
  \item \textbf{Assumptions:} I write the condition that makes the method legal beside the method itself. A correct procedure under the wrong assumptions is still a wrong answer.
  \item \textbf{Representation:} I move freely among words, equations or code, a diagram, and a small numerical example. If one representation disagrees with another, that disagreement is useful evidence.
  \item \textbf{Failure modes:} I keep the nearest counterexample in mind: the boundary where the rule changes, the invariant breaks, or the model stops matching reality.
  \item \textbf{Communication:} My final answer names what the result means and what it does not establish. I do not let notation do the explanatory work for me.
\end{itemize}

\begin{sanitybox}
\textbf{Closed-note check.} Can I teach expectation, conditioning, and generating tools on a blank page, derive or reconstruct its main rule, work a fresh example, and diagnose a deliberately broken solution? If I can only recognize the finished work, I am not done.
\end{sanitybox}

\subsection{How this chapter connects}
I read this chapter as the bridge from functions and transformations of variables to laws of large numbers and central limit theorem. The earlier material supplies the language and constraints; this chapter adds a new reasoning move; the later material asks me to use that move without being told its name. That connection matters because exams and real projects mix chapters on purpose.


\subsection{The full working model}
\begin{itemize}
  \item \textbf{Core claim.} Linearity of expectation needs no independence.
  \item \textbf{What moves.} Conditional expectation is a random variable and projection; total expectation and total variance decompose uncertainty.
  \item \textbf{What keeps the answer honest.} Moment and probability generating functions encode distributions when they exist.
  \item \textbf{Mastery standard.} I can define the objects, rebuild the rule, work a fresh case, compare alternatives, and diagnose a broken solution without relying on recognition.
\end{itemize}

\begin{studenttake}
When a direct joint calculation looks ugly, I condition on the variable that makes the rest simple.
\end{studenttake}

\begin{plainbox}
Linearity of expectation needs no independence.
\end{plainbox}

\begin{examplebox}
\textbf{Question.} If E[X|Y]=2Y and E[Y]=3, find E[X].

\textbf{Walkthrough.} By iterated expectation, E[X]=E[E[X|Y]]=E[2Y]=6.
\end{examplebox}

\begin{warningbox}
The fastest way to get this chapter wrong is to use the visible procedure without naming its assumption, units, time period, reference group, or boundary. I put that condition beside the work and verify the result independently.
\end{warningbox}

\begin{chapterpractice}
\textbf{Practice 1.} If E[X|Y]=2Y and E[Y]=3, find E[X].

\textbf{Solution.} By iterated expectation, E[X]=E[E[X|Y]]=E[2Y]=6.

\medskip
\textbf{Practice 2.} Explain expectation, conditioning, and generating tools to a classmate using one definition, one concrete example, one assumption, and one failure mode.

\textbf{Solution.} Begin with the core claim above, use the worked case as the concrete example, state the condition highlighted by the warning, and choose a failure where that condition does not hold. A complete explanation connects all four pieces instead of listing them.

\medskip
\textbf{Mastery practice A.} Reconstruct expectation, conditioning, and generating tools from a blank page. Include the central definition, the governing rule, one required assumption, and one boundary case.

\textbf{Solution.} Start from the blue idea box, but rewrite it in my own words. Identify the objects and state, name the rule before using it, copy the relevant assumption from the details audit, and choose a boundary where that assumption becomes equality or fails. A complete solution explains why each part belongs; a list of vocabulary is not enough.

\medskip
\textbf{Mastery practice B.} A classmate gets a polished-looking answer to a expectation, conditioning, and generating tools problem but never states a model or checks the result. Write the shortest useful review of that solution.

\textbf{Solution.} I would ask four concrete questions: What exactly are the inputs and desired output? Which rule licenses the main step? Which assumption could invalidate it? Which independent check supports the conclusion? If the answer cannot survive those four questions, I treat the numerical or verbal result as unverified.

\end{chapterpractice}

\section{Laws of large numbers and central limit theorem}

\begin{idea}
The law of large numbers makes sample averages concentrate around expectation; the central limit theorem describes scaled fluctuations around that limit. Assumptions, rate, tail behavior, and finite-sample approximation matter.
\end{idea}
% MASTERCLASS EXPANSION
\subsection{Rebuild the chapter from first principles}
The claim I need to own is this: The law of large numbers makes sample averages concentrate around expectation; the central limit theorem describes scaled fluctuations around that limit. Assumptions, rate, tail behavior, and finite-sample approximation matter. I do not count that as learned just because the sentence looks familiar. I should be able to name the objects, state what changes and what stays fixed, recover the rule from the definitions, and explain why the result is allowed.

\begin{move}
\textbf{My working sequence}
\begin{enumerate}
  \item \textbf{Translate.} Rewrite the prompt in the language of laws of large numbers and central limit theorem. List the known quantities, the unknown, the units or types, and any hidden constraint.
  \item \textbf{Choose.} State the theorem, model, algorithm, accounting rule, or physical law before substituting values. This is where I make the assumption visible.
  \item \textbf{Execute.} Work one justified step at a time and keep intermediate state readable enough that I can audit it later.
  \item \textbf{Verify.} Check a boundary case, sign, unit, invariant, order of magnitude, or alternative derivation. Then translate the result back into an ordinary sentence.
\end{enumerate}
\end{move}

\subsection{Details I would refuse to skip}
\begin{itemize}
  \item \textbf{Definitions:} I should be able to define every technical noun in the chapter without using that same noun in the definition.
  \item \textbf{Assumptions:} I write the condition that makes the method legal beside the method itself. A correct procedure under the wrong assumptions is still a wrong answer.
  \item \textbf{Representation:} I move freely among words, equations or code, a diagram, and a small numerical example. If one representation disagrees with another, that disagreement is useful evidence.
  \item \textbf{Failure modes:} I keep the nearest counterexample in mind: the boundary where the rule changes, the invariant breaks, or the model stops matching reality.
  \item \textbf{Communication:} My final answer names what the result means and what it does not establish. I do not let notation do the explanatory work for me.
\end{itemize}

\begin{sanitybox}
\textbf{Closed-note check.} Can I teach laws of large numbers and central limit theorem on a blank page, derive or reconstruct its main rule, work a fresh example, and diagnose a deliberately broken solution? If I can only recognize the finished work, I am not done.
\end{sanitybox}

\subsection{How this chapter connects}
I read this chapter as the bridge from expectation, conditioning, and generating tools to the cumulative course model. The earlier material supplies the language and constraints; this chapter adds a new reasoning move; the later material asks me to use that move without being told its name. That connection matters because exams and real projects mix chapters on purpose.


\subsection{The full working model}
\begin{itemize}
  \item \textbf{Core claim.} The law of large numbers makes sample averages concentrate around expectation; the central limit theorem describes scaled fluctuations around that limit.
  \item \textbf{What moves.} Assumptions, rate, tail behavior, and finite-sample approximation matter.
  \item \textbf{What keeps the answer honest.} The model becomes useful only when I can apply laws of large numbers and central limit theorem to an unfamiliar case and defend the assumptions.
  \item \textbf{Mastery standard.} I can define the objects, rebuild the rule, work a fresh case, compare alternatives, and diagnose a broken solution without relying on recognition.
\end{itemize}

\begin{studenttake}
LLN tells me where the average goes; CLT tells me the shape of its remaining error.
\end{studenttake}

\begin{plainbox}
The law of large numbers makes sample averages concentrate around expectation; the central limit theorem describes scaled fluctuations around that limit.
\end{plainbox}

\begin{examplebox}
\textbf{Question.} A population has mean 10 and standard deviation 4. Approximate the distribution of the mean of 100 iid observations.

\textbf{Walkthrough.} The sample mean is approximately normal with mean 10 and standard error 4/sqrt(100)=0.4, assuming the CLT approximation is adequate.
\end{examplebox}

\begin{warningbox}
The fastest way to get this chapter wrong is to use the visible procedure without naming its assumption, units, time period, reference group, or boundary. I put that condition beside the work and verify the result independently.
\end{warningbox}

\begin{chapterpractice}
\textbf{Practice 1.} A population has mean 10 and standard deviation 4. Approximate the distribution of the mean of 100 iid observations.

\textbf{Solution.} The sample mean is approximately normal with mean 10 and standard error 4/sqrt(100)=0.4, assuming the CLT approximation is adequate.

\medskip
\textbf{Practice 2.} Explain laws of large numbers and central limit theorem to a classmate using one definition, one concrete example, one assumption, and one failure mode.

\textbf{Solution.} Begin with the core claim above, use the worked case as the concrete example, state the condition highlighted by the warning, and choose a failure where that condition does not hold. A complete explanation connects all four pieces instead of listing them.

\medskip
\textbf{Mastery practice A.} Reconstruct laws of large numbers and central limit theorem from a blank page. Include the central definition, the governing rule, one required assumption, and one boundary case.

\textbf{Solution.} Start from the blue idea box, but rewrite it in my own words. Identify the objects and state, name the rule before using it, copy the relevant assumption from the details audit, and choose a boundary where that assumption becomes equality or fails. A complete solution explains why each part belongs; a list of vocabulary is not enough.

\medskip
\textbf{Mastery practice B.} A classmate gets a polished-looking answer to a laws of large numbers and central limit theorem problem but never states a model or checks the result. Write the shortest useful review of that solution.

\textbf{Solution.} I would ask four concrete questions: What exactly are the inputs and desired output? Which rule licenses the main step? Which assumption could invalidate it? Which independent check supports the conclusion? If the answer cannot survive those four questions, I treat the numerical or verbal result as unverified.

\end{chapterpractice}

\section*{Cumulative study plan}
I use this packet in three passes. First I reconstruct each model and work its examples without looking. Second I mix chapters so the tool is no longer named for me. Third I explain a complete case aloud, including assumptions, units, uncertainty, failure modes, and what evidence would change my conclusion.

\end{document}
