Week 8 adjustments from Robert

Updates Patricia's branch week8:

1. merged main
2. Adjusted file structure and renamed pages. adjusted toc a bit too (titles)
   1. most significant change here is addition of a new page (2nd section) to separate events and random variables. files are now events and variables
3. Adjusted text on front page (same commit as above).
   1. added paragraph at end to point out a few key things (and describe the journey)
   2. probably good enough as-is, but if we have spare time let's revisit it later
4. split page 2 into 2 pages. moved the continuous stuff over to page 3 and outlined more on page 2
5. outlined a case study that we can carry along the pages: discharge in two rivers. this allows connection to the other pages (last year; rest of risk-reliability book)
6. left your last 2 pages mostly as is but added a couple notes as it related to the stuff i foresee myself adding
7. added 1- and 2-random variable pages from last year as sub-pages of the last (new) section

Notes/comments:

1. I added material that i think makes the story better and supports what we will be doing on friday of 1.8; i think it is feasible but we can of course cut cut cut if i don't get my part in.
2. A section somewhere that emphasizes the concept of density would be worth the effort; students can always be reminded about density, especially in higher dimensions.
3. I think it could be useful to include some stuff from last year in week 1.8. See archived version here. makes sense to stick it at the very end if we can add an example that compares the quantitative effect of the multivariate distribution (e.g., calculate and for a few different models, such that the mean/variance of the marginas and correlation coefficient remain constant (change marginals and copula). RIsk/reliability introduction can be left out. Let's discuss.

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