\(\def\rarrow{ \rightarrow}\) \(\def\R{\mathbb{R}}\) \(\def\finv{f^{-1}}\)

Probability

• \(P(A) = u(A)\)
• \(P(A \cup B)\), when \(A,B\) disjoint, is \(u(A)+u(B)\)

• \(P(A|B) = \dfrac{P(A \cap B)}{P(B)}\)

• \(P(B|A) = \dfrac{P(A \cap B)}{P(A)} = \dfrac{P(A|B)*P(B)}{P(A)}\)

GOMO

• Expand sample size of higher up nodes?
  • tried (N.2)2100

Prob Basics

• DEF Probability Space \((\Omega, F, P)\)
  • outcome := a single result from a single experiment run
  • event := set of outcomes
  • \(\Omega\) := set of all outcomes
  • \(F\) := set of all events, a sigma algebra
  • \(P\) Measure on \(P: F \rarrow [0,1]\)

• DEF Measurable function:

  • function between two underlying sets of two measurable spaces that preserve structure of two spaces
  • let \((\Omega, \Sigma)\) \((R,B)\) be measurable spaces Set of outcomes and Set of all events, sigma algebra
  • \(f: \Omega \rarrow R\) is measurable if for every \(E \in B\), \(\finv(E) \in \Sigma\)
  • Another way of looking at
    • \(\sigma(f) \subseteq \Sigma\), the sigma algebra generated by f is contained in Sigma algebra of domain

• DEF Random variable

  • \(X:\Omega \rarrow R\)
  • is a measurable function from probability space to measure space, (called sample space)
  • NOTE I think I need to understand what measurable functions are better to understand what random variables are