Probability

def. An Outcome Space ($\Omega$) is the set of all possible outcomes of an experiment def. An Event ($A$) is a subset of an outcome space. (die face is even)

Example. For a 6-sided die:

$$\begin{array}{rl}\Omega & =\{1,2,3,4,5,6\}\\ A& =\{2,4,6\}\\ A& \subset \Omega \end{array}$$

def. Probability for countable sets. When all outcomes in $\Omega$ are equally likely, and $\Omega$ is a finite set,

$$P(A)=\frac{\#A}{\#\Omega }$$

where $\#A$ denotes the number of elements in set $A$ (its cardinality).

Properties.

• $\mathbb{P}(A^C)=1-\mathbb{P}(A)$
• $\text{If }A\subseteq B\text{ then }P(A)\le P(B)$
• $\text{If }A\subseteq B\text{ then }P(B\cap A^C)=P(B)-P(A)$
• $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ Trivial cases:
• $P(\varnothing )=0$
• $P(\Omega )=1$
• $A\subseteq B$ is equivalent to $A\to B$ (i.e. if A then B)

…for Stochastic Calculus §

Motivation. We need a sigma algebra to define a probability, because there’s problematic set theory problems with uncountably infinite sets.

Sigma-Algebra §

def. Sigma Algebra. A sigma-algebra on a set $\Omega$ denoted $\sigma (\Omega )$ contains certain subsets of $A$, such that it follows the following properties:

1. $\{\varnothing ,\Omega \}\subseteq \sigma (\Omega )\subseteq 2^{\Omega }$ (biggest and smallest possible sigma-algebras)
2. let $A$ be a subset of $\Omega$. If $A\in \sigma (\Omega )$ then $A^C\in \Omega$. (closed under complement)
3. let $A,B$ a subset of $\Omega$. If $A,B\in \sigma (\Omega )$ then $A\cup B\in \sigma (\Omega )$ (closed under finite union) def. Atom of a Sigma Algebra $\mathcal{F}$ is a non-empty set $A\in \mathcal{F}$ where for any other set $B\in \mathcal{F}$, $A\cap B=\varnothing \text{ or }A$.

• All the atoms in $\mathcal{F}$ form a partition on $\Omega$.
• Intuition. The important parts of a sigma-alg is the atoms. Other elements generated by union/complement are there to satisfy measure-theory.

Measure §

def. Measure. let $X,\sigma (X)$. A measure $\mu$ is a function from measure space to to real numbers:

$$\mu :\sigma (X)\to \mathbb{R}$$

that satisfies the following properties:

• Non-negative: $\forall A\in \sigma (X)$, $\mu (A)>0$
• Countable Additivity: $\mu (A\cup B)=\mu (A)+\mu (B)$
• $\mu (\varnothing )=0$ Intuition. Consider a measure as a means to measure the “size” of the set (not Cardinality). On a real number line, the set $(2,5)\equiv \{x\in \mathbb{R}\mid 2<x<5\}$ has length $3$. The set $(5,9)$ has length $4$. Thus $(2,5)\cup (5,9)=(2,9)$ has length $3+4=7$. def. A Measure space is simply a set $X$ and its sigma-algebra $\Sigma$ together in a tuple:

$$(X,\Sigma )$$

Probability Space §

def. A Probability measure on $X,\sigma (X)$ is a special type of measure $P:\sigma (X)\to \mathbb{R}$ that satisfies $P(\Omega )=1$ def. A Probability Space is like a measurable space, but also together with the probability function:

$$(\Omega ,\mathcal{F},\mathbb{P})$$

Interpretations of Probability (Philosophically) §

Motivation. What is probability? Interpretations of its definitions seem meaningless without real-life experiments.

Two main interpretations of probability are:

1. The Objectivist Interpretation—relative frequency of occurrence, if the experiment is conducted indefinitely
2. The Subjectivist Interpretation—degree of belief; how much you would bet on an event.