From ba14b89a35bacaa55daad01c342a0b6b6f0865ad Mon Sep 17 00:00:00 2001
From: Youwen Wu <youwenw@gmail.com>
Date: Thu, 23 Jan 2025 14:21:33 -0800
Subject: [PATCH] auto-update(nvim): 2025-01-23 14:21:33

---
 .../by-course/math-4b/course-notes/main.typ   |  51 ++-
 .../by-course/pstat-120a/course-notes/dvd.typ | 341 ------------------
 .../pstat-120a/course-notes/main.typ          |  66 ++++
 3 files changed, 116 insertions(+), 342 deletions(-)
 delete mode 100644 documents/by-course/pstat-120a/course-notes/dvd.typ

diff --git a/documents/by-course/math-4b/course-notes/main.typ b/documents/by-course/math-4b/course-notes/main.typ
index 354ffb5..320d1fc 100644
--- a/documents/by-course/math-4b/course-notes/main.typ
+++ b/documents/by-course/math-4b/course-notes/main.typ
@@ -250,7 +250,7 @@ When $F(t,y)$ (RHS) is not nice (differentiable) at the initial value $(t_0,
 y_0)$, existence and uniqueness (E/U) could fail. We do have E/U when $F(t,y)$
 is nice.
 
-== Existence of solutions to IVT - linear case
+== Existence of solutions to IVP - linear case
 
 The first order linear ODE is
 
@@ -339,3 +339,52 @@ $ y' = 1 / 2 cos(y) + t, y(0) = -1 $
 
 The righthand side is nice, by @eutheorem, it has a unique solution. We can't
 find an explicit solution.
+
+= Lecture #datetime(day: 23, month: 1, year: 2025).display()
+
+Second order linear differential equation.
+
+$
+  y'' = p(t)y' + q(t)y = g(t)
+$
+
+We say it is homogenous if $g(t) = 0$.
+
+== Initial value problem and uniqueness on 2nd order linear ODEs
+
+Suppose $p(t)$, $q(t)$, and $g(t)$ are given continuous functions defined on
+$I$.
+$
+  y'' = p(t)y' + q(t)y = g(t) \
+  y(t_0) = y_0, y'(t_0) = y'_0
+$
+
+has a unique solution $y(t)$ defined on $I$. Same as first order case. Always write in standard form first before identifying.
+
+#example[
+  What is the largest interval on which the IVP
+  $
+    (t+1)y'' + y = 3, y(0) = 1, y'(0) = 0
+  $
+  is certain to have a solution?
+
+  $
+    y'' + 1 / (t+1) y = 3 / (t+1)
+  $
+
+  So the functions are nice except at -1. Our intervals are
+
+  $
+    (-infinity, -1), (-1, infinity)
+  $
+
+  But we have initial values at 0, so we know for certain that a solution (but not for certain it's the only solution) exists
+
+  $
+    (-1, infinity)
+  $
+]
+
+== 2nd order linear homogenous ODEs
+
+
diff --git a/documents/by-course/pstat-120a/course-notes/dvd.typ b/documents/by-course/pstat-120a/course-notes/dvd.typ
deleted file mode 100644
index 06deb27..0000000
--- a/documents/by-course/pstat-120a/course-notes/dvd.typ
+++ /dev/null
@@ -1,341 +0,0 @@
-#import "@preview/ctheorems:1.1.3": *
-#import "@preview/showybox:2.0.3": showybox
-
-#let colors = (
-  rgb("#9E9E9E"),
-  rgb("#F44336"),
-  rgb("#E91E63"),
-  rgb("#9C27B0"),
-  rgb("#673AB7"),
-  rgb("#3F51B5"),
-  rgb("#2196F3"),
-  rgb("#03A9F4"),
-  rgb("#00BCD4"),
-  rgb("#009688"),
-  rgb("#4CAF50"),
-  rgb("#8BC34A"),
-  rgb("#CDDC39"),
-  rgb("#FFEB3B"),
-  rgb("#FFC107"),
-  rgb("#FF9800"),
-  rgb("#FF5722"),
-  rgb("#795548"),
-  rgb("#9E9E9E"),
-)
-
-#let dvdtyp(
-  title: "",
-  subtitle: "",
-  author: "",
-  abstract: none,
-  bibliography: none,
-  paper-size: "a4",
-  date: "today",
-  body,
-) = {
-  set document(title: title, author: author)
-
-  set std.bibliography(style: "springer-mathphys", title: [References])
-
-  show: thmrules
-
-  set page(
-    numbering: "1",
-    number-align: center,
-    header: locate(loc => {
-      if loc.page() == 1 {
-        return
-      }
-      box(stroke: (bottom: 0.7pt), inset: 0.4em)[#text(
-          font: "New Computer Modern",
-        )[
-          *#author* --- #datetime.today().display("[day] [month repr:long] [year]")
-          #h(1fr)
-          *#title*
-        ]]
-    }),
-    paper: paper-size,
-    // The margins depend on the paper size.
-    margin: (
-      left: (86pt / 216mm) * 100%,
-      right: (86pt / 216mm) * 100%,
-    ),
-  )
-
-
-  set heading(numbering: "1.")
-  show heading: it => {
-    set text(font: "Libertinus Serif")
-
-    block[
-      #if it.numbering != none {
-        text(rgb("#2196F3"), weight: 500)[#sym.section]
-
-        text(rgb("#2196F3"))[#counter(heading).display() ]
-      }
-      #it.body
-      #v(0.5em)
-    ]
-  }
-
-  set text(font: "New Computer Modern", lang: "en")
-
-  show math.equation: set text(weight: 400)
-
-
-  // Title row.
-  align(center)[
-    #set text(font: "Libertinus Serif")
-    #block(text(weight: 700, 26pt, title))
-
-
-    #if subtitle != none [#text(12pt, weight: 500)[#(
-          subtitle
-        )]]
-
-    #if author != none [#text(16pt)[#smallcaps(author)]]
-    #v(1.2em, weak: true)
-
-    #if date == "today" {
-      datetime.today().display("[day] [month repr:long] [year]")
-    } else {
-      date
-    }
-
-  ]
-
-  if abstract != none [
-    #v(2.2em)
-    #set text(font: "Libertinus Serif")
-    #pad(x: 14%, abstract)
-    #v(1em)
-  ]
-
-  set outline(fill: repeat[~.], indent: 1em)
-
-  show outline: set heading(numbering: none)
-  show outline: set par(first-line-indent: 0em)
-
-  show outline.entry.where(level: 1): it => {
-    text(font: "Libertinus Serif", rgb("#2196F3"))[#strong[#it]]
-  }
-  show outline.entry: it => {
-    h(1em)
-    text(font: "Libertinus Serif", rgb("#2196F3"))[#it]
-  }
-
-
-  // Main body.
-  set par(
-    justify: true,
-    spacing: 0.65em,
-    first-line-indent: 2em,
-  )
-
-  body
-
-  // Display the bibliography, if any is given.
-  if bibliography != none {
-    show std.bibliography: set text(footnote-size)
-    show std.bibliography: set block(above: 11pt)
-    show std.bibliography: pad.with(x: 0.5pt)
-    bibliography
-  }
-}
-
-#let thmtitle(t, color: rgb("#000000")) = {
-  return text(
-    font: "Libertinus Serif",
-    weight: "semibold",
-    fill: color,
-  )[#t]
-}
-#let thmname(t, color: rgb("#000000")) = {
-  return text(font: "Libertinus Serif", fill: color)[(#t)]
-}
-
-#let thmtext(t, color: rgb("#000000")) = {
-  let a = t.children
-  if (a.at(0) == [ ]) {
-    a.remove(0)
-  }
-  t = a.join()
-
-  return text(font: "New Computer Modern", fill: color)[#t]
-}
-
-#let thmbase(
-  identifier,
-  head,
-  ..blockargs,
-  supplement: auto,
-  padding: (top: 0.5em, bottom: 0.5em),
-  namefmt: x => [(#x)],
-  titlefmt: strong,
-  bodyfmt: x => x,
-  separator: [. \ ],
-  base: "heading",
-  base_level: none,
-) = {
-  if supplement == auto {
-    supplement = head
-  }
-  let boxfmt(name, number, body, title: auto, ..blockargs_individual) = {
-    if not name == none {
-      name = [ #namefmt(name)]
-    } else {
-      name = []
-    }
-    if title == auto {
-      title = head
-    }
-    if not number == none {
-      title += " " + number
-    }
-    title = titlefmt(title)
-    body = [#pad(top: 2pt, bodyfmt(body))]
-    pad(
-      ..padding,
-      showybox(
-        width: 100%,
-        radius: 0.3em,
-        breakable: true,
-        padding: (top: 0em, bottom: 0em),
-        ..blockargs.named(),
-        ..blockargs_individual.named(),
-        [
-          #title#name#titlefmt(separator)#body
-        ],
-      ),
-    )
-  }
-
-  let auxthmenv = thmenv(
-    identifier,
-    base,
-    base_level,
-    boxfmt,
-  ).with(supplement: supplement)
-
-  return auxthmenv.with(numbering: "1.1")
-}
-
-#let styled-thmbase = thmbase.with(
-  titlefmt: thmtitle,
-  namefmt: thmname,
-  bodyfmt: thmtext,
-)
-
-#let builder-thmbox(color: rgb("#000000"), ..builderargs) = styled-thmbase.with(
-  titlefmt: thmtitle.with(color: color.darken(30%)),
-  bodyfmt: thmtext.with(color: color.darken(70%)),
-  namefmt: thmname.with(color: color.darken(30%)),
-  frame: (
-    body-color: color.lighten(92%),
-    border-color: color.darken(10%),
-    thickness: 1.5pt,
-    inset: 1.2em,
-    radius: 0.3em,
-  ),
-  ..builderargs,
-)
-
-#let builder-thmline(
-  color: rgb("#000000"),
-  ..builderargs,
-) = styled-thmbase.with(
-  titlefmt: thmtitle.with(color: color.darken(30%)),
-  bodyfmt: thmtext.with(color: color.darken(70%)),
-  namefmt: thmname.with(color: color.darken(30%)),
-  frame: (
-    body-color: color.lighten(92%),
-    border-color: color.darken(10%),
-    thickness: (left: 2pt),
-    inset: 1.2em,
-    radius: 0em,
-  ),
-  ..builderargs,
-)
-
-#let problem-style = builder-thmbox(
-  color: colors.at(11),
-  shadow: (offset: (x: 2pt, y: 2pt), color: luma(70%)),
-)
-
-#let exercise = problem-style("item", "Exercise")
-#let problem = exercise
-
-#let theorem-style = builder-thmbox(
-  color: colors.at(6),
-  shadow: (offset: (x: 3pt, y: 3pt), color: luma(70%)),
-)
-
-#let example-style = builder-thmbox(
-  color: colors.at(16),
-  shadow: (offset: (x: 3pt, y: 3pt), color: luma(70%)),
-)
-
-#let theorem = theorem-style("item", "Theorem")
-#let lemma = theorem-style("item", "Lemma")
-#let corollary = theorem-style("item", "Corollary")
-
-#let definition-style = builder-thmline(color: colors.at(8))
-
-// #let definition = definition-style("definition", "Definition")
-#let proposition = definition-style("item", "Proposition")
-#let remark = definition-style("item", "Remark")
-#let observation = definition-style("item", "Observation")
-
-// #let example-style = builder-thmline(color: colors.at(16))
-
-#let example = example-style("item", "Example")
-
-#let proof(body, name: none) = {
-  v(0.5em)
-  [_Proof_]
-  if name != none {
-    [ #thmname[#name]]
-  }
-  [.]
-  body
-  h(1fr)
-
-  // Add a word-joiner so that the proof square and the last word before the
-  // 1fr spacing are kept together.
-  sym.wj
-
-  // Add a non-breaking space to ensure a minimum amount of space between the
-  // text and the proof square.
-  sym.space.nobreak
-
-  $square.stroked$
-  v(0.5em)
-}
-
-#let fact = thmplain(
-  "item",
-  "Fact",
-  titlefmt: content => [*#content.*],
-  namefmt: content => [_(#content)._],
-  separator: [],
-  inset: 0pt,
-  padding: (bottom: 0.5em, top: 0.5em),
-)
-#let abuse = thmplain(
-  "item",
-  "Abuse of Notation",
-  titlefmt: content => [*#content.*],
-  namefmt: content => [_(#content)._],
-  separator: [],
-  inset: 0pt,
-  padding: (bottom: 0.5em, top: 0.5em),
-)
-#let definition = thmplain(
-  "item",
-  "Definition",
-  titlefmt: content => [*#content.*],
-  namefmt: content => [_(#content)._],
-  separator: [],
-  inset: 0pt,
-  padding: (bottom: 0.5em, top: 0.5em),
-)
diff --git a/documents/by-course/pstat-120a/course-notes/main.typ b/documents/by-course/pstat-120a/course-notes/main.typ
index 0d26462..71fe880 100644
--- a/documents/by-course/pstat-120a/course-notes/main.typ
+++ b/documents/by-course/pstat-120a/course-notes/main.typ
@@ -774,4 +774,70 @@ us generalize to more than two colors.
 
 = Discussion section #datetime(day: 22, month: 1, year: 2025).display()
 
+= Lecture #datetime(day: 23, month: 1, year: 2025).display()
 
+== Independence
+#definition("Independence")[
+  Two events $A subset Omega$ and $B subset Omega$ are independent if and only if
+  $ P(B sect A) = P(B)P(A) $
+  "Joint probability is equal to product of their marginal probabilities."
+]
+
+#fact[This definition must be used to show the independence of two events.]
+
+#fact[
+  If $A$ and $B$ are independent, then,
+  $
+    P(A | B) = underbrace((P(A sect B)) / P(B), "conditional probability") = (P(A) P(B)) / P(B) = P(A)
+  $
+]
+
+#example[
+  Flip a fair coin 3 times. Let the events:
+
+  - $A$ = we have exactly one tails among the first 2 flips
+  - $B$ = we have exactly one tails among the last 2 flips
+  - $D$ = we get exactly one tails among all 3 flip
+
+  Show that $A$ and $B$ are independent.
+  What about $B$ and $D$?
+
+  Compute all of the possible events, then we see that
+
+  $
+    P(A sect B) = (hash (A sect B)) / (hash Omega) = 2 / 8 = 4 / 8 dot 4 / 8 = P(A) P(B)
+  $
+
+  So they are independent.
+
+  Repeat the same reasoning for $B$ and $D$, we see that they are not independent.
+]
+
+#example[
+  Suppose we have 4 red and 7 green balls in an urn. We choose two balls with replacement. Let
+
+  - $A$ = the first ball is red
+  - $B$ = the second ball is greeen
+
+  Are $A$ and $B$ independent?
+
+  $
+    hash Omega = 11 times 11 = 121 \
+    hash A = 4 dot 11 = 44 \
+    hash B = 11 dot 7 = 77 \
+    hash (A sect B) = 4 dot 7 = 28
+  $
+]
+
+#definition[
+  Events $A_1, ..., A_n$ are independent (mutually independent) if for every collection $A_i_1, ..., A_i_k$, where $2 <= k <= n$ and $1 <= i_1 < i_2 < dots.c < i_k <= n$,
+
+  $
+    P(A_i_1 sect A_i_2 sect dots.c sect A_i_k) = P(A_i_1) P(A_i_2) dots.c P(A_i_k)
+  $
+]
+
+#definition[
+  We say that the events $A_1, ..., A_n$ are *pairwise independent* if any two
+  different events $A_i$ and $A_j$ are independent for any $i != j$.
+]