more stuff

book/_toc.yml

@@ -95,9 +95,21 @@ parts:
       title: Multivariate Distributions
       sections:
       - file: multivariate/events
+        title: Dependent Events
       - file: multivariate/variables
+        title: Continuous Variables
       - file: multivariate/gaussian
+        title: Multivariate Gaussian
       - file: multivariate/nongaussian
+        title: Non-Gaussian Distributions
+      - file: multivariate/functions
+        title: Functions of Random Variables
+        sections:
+        - file: rr/prob-design/one-random-variable
+        - file: rr/prob-design/two-random-variables
+        - file: rr/prob-design/exercise
+          sections:
+          - file: rr/prob-design/exercise_solutions
 
 #   - caption: Q2 Topics
 #     numbered: 2

book/multivariate/events.md

+(multivariate_events)=
+# Events: Fundamentally Refreshing
 
-# Basic concepts to start
+Before going further into _continuous multivariate distributions,_ we will start with a reminder of some basic concepts you have seen previously: independence and conditional probability, illustrated by considering the probability of _events._
 
-Before going further into continuous multivariate distributions, we will start with a reminder of some basic concepts you have seen previously: independence, AND and OR probabilities, and conditional probability.
+```{admonition} Event
+:class: tip
+
+In probability theory, an _event_ is considered to be the outcome of an experiment with a specific probability. 
+
+**some other stuff**
 
 If you also need further practice or to revisit other concepts such as mutually exclusive events or collectively exhaustive, you can go [here](https://teachbooks.github.io/learn-probability/section_01/Must_know_%20probability_concepts.html).
+```
+
+## Discrete Events
+
+As we are working towards multivariate _continuous_ distributions we will these events will be referred to _discrete_ events to distinguish them from.
+
+In this case our sample space is:
+- still 1
+- each event is a random variable
+- to facilitate the venn diagram and "event-based" analogies we will only consider binary cases for each event, so $\leq$ and $>$ cases (can illustrate for more than binary cases, but why bother)
+- 
 
-## AND and OR probabilities: Venn diagrams
+Great. Now let's review a few key concepts (quickly!).
 
-Let's move back to discrete events to explain what AND and OR probabilities are. Imagine two events, A and B. These can be, for instance, the fact that it rains today (A) and the fact that the temperature is below 10 degrees (B). Each of these events will have a probability of ocurring, denoted here as $P(A)$ and $P(B)$, respectively.
+**idea**: simply list condition, total probability, independence rule, Bayes rule (lays out terms and usage), then the following sections briefly illustrate these things with the flood example and Venn diagrams.
 
+## Case Study
 
-```{figure} ../venn-events.png
+Imagine two events, A and B:
+- A represents river A flooding
+- B represents river B flooding
+
+Each of these events will have a probability of ocurring, denoted here as $P(A)$ and $P(B)$, respectively.
+
+
+```{figure} ./figures/venn-events.png
 
 ---
 
@@ -20,7 +46,7 @@ Venn diagram of the events A and B.
 
 The AND probility or intersection of the events A and B, $P(A \cap B)$, is defined as the probability that both events happen at the same time and, thus, it would be represented in our diagram as shown in the figure below.
 
-```{figure} ../venn-intersection.png
+```{figure} ./figures/venn-intersection.png
 
 ---
 
@@ -28,6 +54,16 @@ The AND probility or intersection of the events A and B, $P(A \cap B)$, is defin
 Venn diagram of the events A and B, and AND probability.
 ```
 
+Thus there are two ways we have of describing the same probability:
+- intersection
+- AND
+
+we will use these interchangeably.
+
+Keep an eye out for related (English) words: both, together, joint, ...
+
+### 
+
 The OR probability or union of the events A and B, $P(A \cup B)$, is defined as the probability that either one of the two events happen or both of them. This probability can be computed as 
 
 $$
@@ -36,25 +72,14 @@ $$
 
 This is, we add the probabilities of occurrence of the event A and B and we deduct the intersection of them, to avoid counting them twice.
 
-## AND and OR probabilities from samples
-
-
 
-## Independence
 
-When two random variables, X and Y, are independent, it means that the occurrence or value of one variable does not influence the occurrence or value of the other variable.
 
-Formally, X and Y are considered independent **if and only if** the joint probability function (or cumulative distribution function) can be factorized into the product of their marginal probability functions (or cumulative distribution functions). This is, 
-
-$F(x, y) = P(x<X \bigcap y<Y ) = P(x<X)P(y<Y) = F(x)F(y)$
-
-The different relationships above highlights the connection between the joint cumulative distribution function (CDF) and the marginal CDFs of two independent random variables, X and Y.
-
-Definition of independence 
-
-Definition of And and OR probabilities using Venn diagrams
+## AND and OR probabilities from samples
 
-Move to continuous distributions and compute them from samples
+**we can illustrate samples in the venn diagram as dots with labels:
+- simple counting exercises will illustrate the probabilities
+- 
 
 ## Conditional probability
 

book/multivariate/functions.md

+(multivar_functions)=
+# Functions of Random Variables
+
+In contrast to earlier weeks, this is where there is dependence between the input random variables.
+
+binary cases are **boring**
+
+Up to this point we have only done binary cases with two RV's (both or either river flooding). In other words, when the problem of interest is based on some set of A and B. What about when this is described as a continuous function? Then we have a (familiar) function of random variables! We will illustrate this case in the next section. **this is where the 1- and 2-random variable pages from 2023 can be included.
+
+**1- and 2-random variable cases from 2023.**
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book/multivariate/gaussian.md

@@ -5,4 +5,6 @@ Definition of bivariate Gaussian
 
 Move to 3D
 
-Analytical conditionalization of the 3D Gaussian: 2D margin!
\ No newline at end of file
+Analytical conditionalization of the 3D Gaussian: 2D margin!
+
+**case study**: return to the river flooding case and illustrate the effect of dependence. figure and table.
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book/multivariate/overview.md

@@ -16,8 +16,24 @@ $$
 \rho(X_1,X_2)=\frac{Cov(X_1,X_2)}{\sigma_{X_1} \sigma_{X_2}} 
 $$ 
 
-where $X_1$ and $X_2$ are random variables, $Cov(X_1,X_2)$ is their covariance, and $\sigma_{X_1}$ and $\sigma_{X_2}$ are the standard deviations of $X_1$ and $X_2$. $-1 \leq \rho(X_1,X_2) \leq 1$, being $\rho(X_1,X_2)=-1$ a perfect negative linear correlation, and $\rho(X_1,X_2)=1$ a perfect positive linear correlation. If $\rho(X_1,X_2) \to 0$, we say that $X_1$ and $X_2$ are independent[^note]. This is, that having information about $X_1$ does not provide us with information about $X_2$. The interactive element below allows you to play around with the correlation value yourself. Observe how the distribution's density contours, or the scattered data, changes when you adapt the correlation value.
+where $X_1$ and $X_2$ are random variables, $Cov(X_1,X_2)$ is their covariance, and $\sigma_{X_1}$ and $\sigma_{X_2}$ are the standard deviations of $X_1$ and $X_2$. $-1 \leq \rho(X_1,X_2) \leq 1$, being $\rho(X_1,X_2)=-1$ a perfect negative linear correlation, and $\rho(X_1,X_2)=1$ a perfect positive linear correlation. If $\rho(X_1,X_2) \to 0$, we say that $X_1$ and $X_2$ are independent[^note]. This is, that having information about $X_1$ does not provide us with information about $X_2$. The interactive element below allows you to play around with the correlation value yourself. Observe how the distribution's _density_ contours, or a scatter plot of _samples,_ change when you adjust the correlation.
 
 <iframe src="../_static/elements/element_correlation.html" width="600" height="400" frameborder="0"></iframe>
 
+**Overview of this Chapter**
+
+Our ultimate goal is to construct and validate a model to quantify probability for combinations of more than one random variable of interest (i.e., to quantify various types of uncertainty). Specifically, 
+
+$$
+f_X(x) \;\; \textrm{and} \;\; F_X(x)
+$$
+
+where $X$ is a vector of continuous random variables and $f$ and $F$ are the multivariate probability density function (PDF) and cumulative distribution functions (CDF), respectively. Often we will use _bivariate_ situations (two random variables) to illustrate key concepts, for example:
+
+$$
+f_{X_1,X_2}(x_1,x_2) \;\; \textrm{and} \;\; F_{X_1,X_2}(x_1,x_2)
+$$
+
+This chapter begins with a refresher on some fundamental aspects of probability theory that are typically covered in BSc courses on the subject, for example, dependence/independence, probability of binary events and conditional probability. Using the _bivariate_ paradigm, we will build a foundation on which to apply the multivariate Gaussian distribution (introduced in earlier chapters), as well as introduce alternative _multivariate distributions._ The chapter ends with a brief introduction to _copulas_ as a straightforward approach for evaluating two random variables that are dependent _and_ described by non-Gaussian marginal distributions (the previous chapter).
+
 [^note]: That is an intuitive definition of independence. For a more formal definition of independence, visit the next page of the chapter.
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book/multivariate/variables.md

+(multivariate_variables)=
+# Multivariate Random Variables
+
+
+
+
+
+
+
+## Independence
+
+When two random variables, X and Y, are independent, it means that the occurrence or value of one variable does not influence the occurrence or value of the other variable.
+
+Formally, X and Y are considered independent **if and only if** the joint probability function (or cumulative distribution function) can be factorized into the product of their marginal probability functions (or cumulative distribution functions). This is, 
+
+$F(x, y) = P(x<X \bigcap y<Y ) = P(x<X)P(y<Y) = F(x)F(y)$
+
+The different relationships above highlights the connection between the joint cumulative distribution function (CDF) and the marginal CDFs of two independent random variables, X and Y.
+
+Definition of independence 
+
+### Illustration
+
+Definition of And and OR probabilities using bivariate plots and compared to the Venn diagrams.
+
+## Samples
+
+Compute probabilities from samples.
+
+Illustrate the difference between the theoretical and empirical probabilities. Include a table that summarizes them and describe how this can be used to validate the multivariate distribution (**obviously** we should illustrate a case where dependence is important: many observations where _both_ rivers flood).
+
+**Illustrate explicitly that this is the thing that is inaccurate in the example:**
+
+$$
+F_{X_1,X_2}(X_1>x_1,X_2>x_2)
+$$
+
+**So now we need a way to describe dependence!** 
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