Probability space

$\Omega$ sample space

Machinery
$\beta\equiv \sigma$ - field on $\Omega$
($\beta$ is a subset of $\Omega$ and has special properties)
Set Defs

• $\Omega \in \beta$
• if event $E=\{w: w \in A_E \} \in \beta$ then so is $\bar{E} \;(E^c)$
  A_E is event set
• if $E_i \in \beta \forall i \rightarrow \bigcup_{i=1}^{\infty} E_i \in \beta$

Measure Defs
Probability measure P

1.

$0\le P(E) \le 1 \qquad \forall E\in\beta$

2.

$P(\Omega)=1$

3.

With $E_i\bigcap E_j=\phi \;\forall i\ne j \rightarrow P(\bigcup_{i=1}^{\infty})=\sum_{i=1}^{\infty} P(E_i)$
ie. probs. of disjoint events add

Fallout

1.

$P(\bar{E})=1-P(E)$

2.

$P(\phi)=0$

3.

$P(E_1\bigcup E_2)=P(E_1)+P(E_2)-P(E_1\bigcap E_2)$
growup notation
P(E₁ + E₂)=P(E₁)+P(E₂)-P(E₁E₂)

4.

if $E_1 \subset E_2$ then $P(E_2)\ge P (E_1)$

conditional prob
$P(E_1\vert E_2)= \left\{\begin{array}{ll} \displaystyle {\frac{P(E_1 E_2)}{P(... ...mbox{provided} E_2 \ne 0 } \\ \displaystyle {0} & {E_2=0} \end{array}\right.$
If E₁ and E₂ indept. above implies P(E₁ E₂)=P(E₁)P(E₂)

Partition
$\{E_i\}^n_{i=1}$ is a finite partition of sample space

$$\sum_{i=1}^n E_i=\Omega \qquad E_i E_j=\phi\;\;i\ne j \;\; i,j=1,2,...,n$$

can project on outcome A onto E_i!

$$P(A)=\sum_{i=1}^n P(E_i)P(A\vert E_i)$$

known as total probability theorem, which leads to

$$\begin{eqnarray*}P(E_i\vert A)&=&\frac{P(E_i)P(A\vert E_i)}{P(A)}\\ &=&\frac{P(E_i)P(A\vert E_i)}{\sum_{i=1}^n P(E_i)P(A\vert E_i)} \end{eqnarray*}$$

which is known as Bayes Thm.