auto-update(nvim): 2025-01-08 20:40:52

documents/by-course/pstat-120a/course-notes/main.typ

@@ -58,7 +58,7 @@ With arbitrary sets $A$, $B$:
 + $a in.not A$ ($a$ is not a member of the set $A$)
 + $A subset.eq B$ (Set theory: $A$ is a subset of $B$) (Stats: $A$ is a sample space in $B$)
 + $A subset B$ (Proper subset: $A != B$)
-+ $A^c$ or $A'$ (read "complement of $A$", and gives all the elements in the universal set not in $A$)
++ $A^c$ or $A'$ (read "complement of $A$," and introduced later)
 + $A union B$ (Union of $A$ and $B$. Gives a set with both the elements of $A$ and $B$)
 + $A sect B$ (Intersection of $A$ and $B$. Gives a set consisting of the elements in *both* $A$ and $B$)
 + $A \\ B$ (Set difference. The set of all elements of $A$ that are not also in $B$)
@@ -71,15 +71,24 @@ We can also write a few of these operations precisely as set comprehensions.
 + $A sect B = {x | x in A and x in B}$ (here $and$ is the logical AND)
 + $A \\ B = {a | a in A and a in.not B}$
 + $A times B = {(a,b) | forall a in A, forall b in B}$
-+ $A' = A sect Omega$, where $Omega$ is the _universal set_.
+
+Take a moment and convince yourself that these definitions are equivalent to
+the previous ones.
 
 #definition[
   The universal set $Omega$ is the set of all objects in a given set
   theoretical universe.
 ]
 
-Take a moment and convince yourself that these definitions are equivalent to
-the previous ones.
+With the above definition, we can now introduce the set complement.
+
+#definition[
+  The set complement $A'$ is given by
+  $
+    A' = Omega \\ A
+  $
+  where $Omega$ is the _universal set_.
+]
 
 #example[The real plane][
   The real plane $RR^2$ can be defined as a Cartesian product of $RR$ with
@@ -112,7 +121,8 @@ as the notation for $n$ dimensional spaces in $RR$?
   Let $N(A)$ be the number of elements in $A$. $N(A)$ is called the _cardinality_ of $A$.
 ]
 
-Sets are either finite or infinite. Finite sets have a fixed finite cardinality.
+We say a set is finite if it has finite cardinality, or infinite if it has an
+infinite cardinality.
 
 Infinite sets can be either _countably infinite_ or _uncountably infinite_.
 
@@ -198,7 +208,7 @@ This gives us the following equivalent statement:
   $ A = {(4,6,), (5,5),(6,4)} $
 ]
 
-Set theory terms $<-> $ probability terms:
+Probabilistic concepts in the parlance of set theory:
 
 - Superset ($Omega$) $<->$ sample space
 - Element $<->$ outcome / sample point ($omega$)
@@ -237,14 +247,19 @@ $
 
 == Subjective approach
 
-Personal definition of probability.
+Personal definition of probability. Not "real" probability, merely co-opting
+its parlance to lend credibility to subjective judgements of confidence.
 
 == Axiomatic approach
 
-Our focus.
+Our focus in PSTAT 120A. It seems rather silly to call this approach axiomatic
+given we are essentially just defining a function with a few given properties
+and deriving theorems from it while working atop our pre-existing (shaky,
+non-rigorous) "axioms" of set theory, but this is the terminology that the
+course uses.
 
 #definition[
-  $P(dot)$ is a set function satisfying the 3 axioms
+  Let $P : X -> RR$ be a function satisfying the following axioms (properties).
 
   + $P(A) >= 0, forall A$
   + $P(Omega) = 1$
@@ -252,6 +267,8 @@ Our focus.
     $ P(union.big_(i=1)^infinity A_i) = sum_(i=1)^infinity P(A_i) $
 ]
 
+Now let us show various results with $P$.
+
 #proposition[
   $ P(emptyset) = 0 $
 ]
@@ -271,8 +288,17 @@ Our focus.
   $ P(union.big^n_(i=1) A_i) = sum^n_(i= 1) P(A_i) $
 ]
 
+This is mostly a formal manipulation to derive the obviously true proposition from our axioms.
+
 #proof[
-  Consider $(A_1, A_2, ..., A_n, emptyset, emptyset, ...)$.
+  Write any finite set $(A_1, A_2, ..., A_n)$ as an infinite set $(A_1, A_2, ..., A_n, emptyset, emptyset, ...)$. Then
+  $
+    P(union.big_(i=1)^infinity A_i) = sum^n_(i=1) P(A_i) + sum^infinity_(i=n+1) P(emptyset) = sum^n_(i=1) P(A_i)
+  $
+  And because all of the elements after $A_n$ are $emptyset$, their union adds no additional elements to the resultant union set of all $A_i$, so
+  $
+    P(union.big_(i=1)^infinity A_i) = P(union.big_(i=1)^n A_i) = sum_(i=1)^n P(A_i)
+  $
 ]
 
 #proposition[Complement][
@@ -284,7 +310,8 @@ Our focus.
     A' union A &= Omega \
     A' sect A &= emptyset \
     P(A' union A) &= P(A') + P(A) &"(by axiom 3)"\
-    = P(Omega) &= 1 &"(by axiom 2)"
+    = P(Omega) &= 1 &"(by axiom 2)" \
+    therefore P(A') &= 1 - P(A)
   $
 ]
 
@@ -318,6 +345,10 @@ Our focus.
   $
 ]
 
+#remark[
+  This is a stronger result of axiom 3, which generalizes for all sets $A$ and $B$ regardless of whether they're disjoint.
+]
+
 #example[
   Select one card from a deck of 52 cards.
 
@@ -381,9 +412,7 @@ $
   $
 ]
 
-#example[
-  Birthdays.
-
+#example[Birthdays][
   What is the probability two people share the same birthday?
 
   $