Lecture 001

Definitions

Sample Space ($\Omega$): set of all possible outcome

• Discrete Sample Space: number of outcome countable (finite or countably infinite)

• Continuous Sample Space: number of outcome uncountable

Mutually Exclusive (joint): two events are mutually exclusive if $E_1 \cap E_2 = \emptyset$

• therefore $Pr\{E_1 \cup E_2\} = Pr\{E_1\} + Pr\{E_2\}$

• therefore $Pr\{E_1 \cap E_2\} = 0$

• therefore $E_1, E_2$ are dependent

Partition: events $E_1, ..., E_n$ partition set $F$ iff $E_1, ..., E_n$ forms a partition of $F$

Event ($E$): a subset of sample space $\Omega$. An event can either happen or not happen ($E$ and $\cap{E}$ are mutually exclusive)

$Pr\{E\}$ the probability outcome of experiment is in set $E$

1. Non-negative: $(\forall E)(Pr\{E\} \geq 0)$
2. Additive: if $E_1, ..., E_n$ mutually exclusive, then $Pr\{$E_1 \cup ... \cup E_n$\} = Pr\{E_1\} + ... + Pr\{E_n\}$
3. Normalization: $Pr\{\Omega\} = 1$

Note: events $E_1, ..., E_n$ should be countable many

Calculation

Intersecting Event: $Pr\{E \cup F\} = Pr\{E\} + Pr\{F\} - Pr\{E \cap F\}$

Union Bound: $Pr\{E \cup F\} \leq Pr\{E\} + Pr\{F\}$

Union Bound (and inverting probability, of $\forall, \exists$ quantifier) is very useful when you have $\max(\cdot)$ or $\min(\cdot)$

Conditional Probability of event $E$ given $F$: assume $Pr\{F\} > 0$, $Pr\{E | F\} = \frac{Pr\{E \cap F\}}{Pr\{F\}}$

Chainrule: $Pr\{\cup_{i = 1}^n E_i\} = Pr\{E_1\} + Pr\{E_2 | E_1\} + Pr\{E_3 | E_1 \cup E_2\} ... Pr\{E_n | E_1 \cup E_2, ..., \cup E_{n-1}\}$ $Pr\{E \cap F\} = Pr\{F\} \cdot Pr\{E | F\}$: think we are in $F$ first, and then we choose $Pr\{E | F\}$

Independent Event: $E \perp F$ two definitions are the same

• Definition 1: if $Pr\{E \cap F\} = Pr\{E\} \cdot Pr\{F\}$

• Definition 2: if $Pr\{E | F\} = Pr\{E\}$ when $Pr\{F\} > 0$

Proof: $E \perp G \implies E \perp \bar{G}$

\begin{align*} Pr\{E \cap \bar{G}\} & = Pr\{E - (E \cap G)\} \\ & = Pr\{E\} - Pr\{E \cap G\} \\ & = Pr\{E\} - Pr\{E\} \cdot Pr\{G\} \\ & = Pr\{E\} (1 - Pr\{G\}) \\ & = Pr\{E\} \cdot Pr\{\bar{G}\} \\ \end{align*}

Multiple Independent Event:

• Wrong definition: $Pr\{E \cap F \cap G\} = Pr\{E\} \cdot Pr\{F\} \cdot Pr\{G\}$

• Wrong definition: Pair-wise Independent, 3-way Independent

• Good definition: Event $A_1, A_2, ..., A_n$ are Full Independent if $(\forall S \subseteq \{1, 2, ..., n\})(Pr\{\bigcap_{i \in S} A_i\} = \prod_{i \in S}Pr\{A_i\})$

Pairwise Independence: $(\forall i, j)(Pr\{A_i \cap A_j\} = Pr\{A_i\} \cdot Pr\{A_j\})$

Conditional Independent: $Pr\{E \cap F | G\} \neq Pr\{E | G\} \cdot Pr\{F | G\}$

• Independence does not imply Conditional Independence: let $E$ be "1st coin is head", let $F$ be "2nd coin is head", let $G$ be "both coins are the same"

• Conditional Independence does not imply Independence

Law of Total Probability: $E = (E \cap F) \cup (E \cap \bar{F})$ ($Pr\{E\} = Pr\{E | F\} \cdot Pr\{F\} + Pr\{E | \bar{F}\} \cdot Pr\{\bar{F}\}$)

General Law of Total Probability: for $F_1, F_2, ..., F_n$ partition the whole space $\Omega$, then

\begin{align*} Pr\{E\} &= \sum_{i = 1}^n Pr\{E \cup F_i\}\\ &= \sum_{i = 1}^n Pr\{E | F_i\} \cdot Pr\{F_i\}\\ &= Pr\{E | F_1\} \cdot Pr\{F_1\} + Pr\{E | F_2\} \cdot Pr\{F_2\}+ ... + Pr\{E | F_n\} \cdot Pr\{F_n\}\\ \end{align*}

Conditional Law of Total Probability: for $F_1, F_2, ..., F_n$ partition the whole space $\Omega$, then

\begin{align*} Pr\{E | G\} =& \sum_{i = 1}^n Pr\{E | F_i \cap G\} \cdot Pr\{F_i | G\}\\ =& Pr\{E | F_1 \cap G\} \cdot Pr\{F_1 | G\} + Pr\{E | F_2 \cap G\} \cdot Pr\{F_2 | G\}+ ... + Pr\{E | F_n \cap G\} \cdot Pr\{F_n | G\}\\ \end{align*}

Bayes Law:

Pr\{F | E\} = \frac{Pr\{F \cap E\}}{Pr\{E\}} = \frac{Pr\{F\} \cdot Pr\{E | F\}}{Pr\{E\}} = \frac{Pr\{F\} \cdot Pr\{E | F\}}{Pr\{E | F\} \cdot Pr\{F\} + Pr\{E | \bar{F}\} \cdot Pr\{\bar{F}\}}

Extended Bayes Law: Suppose $F_1, F_2, ..., F_n$ partition $\Omega$, then $Pr\{F | E\} = \frac{Pr\{F\} \cdot Pr\{E | F\}}{\sum_{j = 1}^n Pr\{E | F_j\} \cdot Pr\{F_j\}}$

Random Variable: a mapping from experiment outcome to number we care about

• random variable is a mapping

• but once the mapping's outcome is set to specific value, it is an event

Example:

• Experiment roll 2 dies

• Outcome: (1, 4)

• Random Variable: the larger of 2 rolls

• $X = 4$: event

• $X < 4$: event