From 5b594852070434278c5778abcef4409d3690a55b Mon Sep 17 00:00:00 2001
From: Holden Rohrer <hr@hrhr.dev>
Date: Fri, 21 Aug 2020 17:31:51 -0400
Subject: more lectures

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+Convention:
+N = {0,1,2,...}
+N* = {1,2,3,...}
+
+HW problems 1.11 #1-4 (in book)
+
+The unit square, with #Omega = \infty, has probability zero for lines.
+
+Developing a single framework for discrete {heads, tails} and picking a
+point from [0,1].
+
+Ex:
+
+Probability of getting a prime number choosing at random from N*
+
+Def: Let Omega be a sample space. An event space E is a non-empty
+collection of subsets of Omega such that:
+    if A \in E, A^c \in E
+        Lemma Def: iff B \in S and B \not\in A, B \in A^c
+    if A_1,A_2,...,A_n \in E, then the union is in E.
+
+Def: An event is called *elementary* if its cardinality is 1
+
+In general, the power set of Omega (2^Omega = \bb P(Omega)) is the
+collection of all subsets of omega. (sometimes called complete/total
+event space)
+
+Trivial event space: {empty, Omega}
+
+Take Omega = {1,..,7}.
+E_1 = 2^Omega. #E_1 = 2^#Omega = 2^7.
+E_2 = { {3}, {1,2,4,5,6,7}, empty, Omega}. This is also a valid event sp
+
+In general, regardless of cardinality of Omega, we will take E = 2^Omega
+because all event spaces are subsets of the power set.
+
+N, N*, Q, Z, ...
+are infinite countable sets. There is a bijection between these sets and
+N.
+
+R is an infinite uncountable set. Rigorously defining Omega = R is very
+difficult. But this is outside the scope of this course.
+
+Def: A probability measure is a set function, usually denoted by P, from
+E to [0,+\infty] = R^+
+
+(i) P(Omega) = 1.
+(ii) P(Union of A_1...A_n) = \sum_i=1^n P(A_i) if A_1..A_n is pairwise
+disjoint.
+
+Def: Pairwise disjoint if A_i intersect A_j = empty for all i != j.
+
+Function:
+    P: E -> [0,+\infty)
+        For all A in E, there exists a unique P(A)
+
+Upside down A = for all
+Backwards E = there exists
+Backwards E ! = there exists a unique
+
+Review set operations, i.e. union, intersection, complementation,
+commutativity, distributivity, associativity
+
+For Omega finite or infinite,
+
+\sum_\Omega P({\omega_i}) = P(\Omega) = 1.
+P({\omega_i}) \geq 0.
+
+Uniform probabliity:
+    P({\omega_i}) = p_i = 1/#\Omega
+    Each element is equally likely.
+    If A \subset \Omega, P(A) = #A/#Omega
-- 
cgit