2 years ago

# Probability and Statistics

#### Lecture-5

From the relative frequency definition we can observe three basic properties of chance,that are
i). $\frac{f_{n}(E)}{n}\ge0~\forall~E\Rightarrow P(E)=\lim_{n \to \infty}\frac{f_{n}(E)}{n}\ge0 \leftarrow$ Non-negativity property
ii). $f_{n}(\Omega)=n\Rightarrow P(\Omega)=\lim_{n \to \infty}\frac{f_{n}(\Omega)}{n}=\lim_{n \to \infty}1=1 \leftarrow$Certainty property
iii). If $E_{1}\cap E_{2}=\emptyset$ then $f_{n}(E_{1}\cup E_{2})=f_{n}(E_{1})+f_{n}(E_{2})$
then $P(E_{1}\cup E_{2})=\lim_{n \to \infty} \frac{f_{n}(E_{1}\cup E_{2})}{n}=\lim_{n \to \infty} \frac{f_{n}(E_{1})+f_{n}(E_{2})}{n}=\lim_{n \to \infty} \frac{f_{n}(E_{1})}{n} + \lim_{n \to \infty} \frac{f_{n}(E_{2})}{n}=P(E_{1})+P(E_{2})$
$\therefore P(E_{1}\cup E_{2})=P(E_{1})+P(E_{2}) \leftarrow$ Additive property

Axiomatic definition of probability:
Let $(\Omega,\mathscr{F})$ be a measurable space, a set function $P:\mathscr{F}\rightarrow \mathbb{R}$ is called a probability(measure) function on $(\Omega,\mathscr{F})$ if it satisfies the following axioms.

Axiom 1:$P(E)\ge0$ $\forall E\in\mathscr{F}$ (Non-negativity axiom)
Axiom 2:$P(\Omega)=1$ (Certainty axiom)
Axiom 3: For any sequence of events $E_{1},E_{2},E_{3},...,E_{n}....$ in $\mathscr{F}$ such that $E_{i}\cap E_{j}=\emptyset$ for $i\ne j$ then,
$P(\bigcup_{i=1}^{\infty}E_{i})=\sum_{i=1}^{\infty}P(E_{i})$ (Countable additivity axiom)

Here the triple $(\Omega,\mathscr{F},P)$ is called a probability space(or probability model)

Let $I$ be an index set,

• Collection of events $\{E_{i}\}_{i\in I}$ in $\mathscr{F}$, are said to be mutually exclusive if $E_{i}\cap E_{j}=\emptyset$ for $i \ne j$ (At most on event in collection can occur when we performs an experiment which means no two events in the collection can happen simultaneously)
• Collection of events $\{E_{i}\}_{i\in I}$ in $\mathscr{F}$, are said to be exhaustive if $\bigcup_{i\in I}E_{i}=\Omega$ (At least one event in collection always occur when we performs an experiment)
• Collection of events $\{E_{i}\}_{i\in I}$ in $\mathscr{F}$, are said to form a partition for $\Omega$ if they are both mutually exclusive and exhaustive (Exactly one event in collection can occur when we performs an experiment)

In all of the following theorems we consider probability space $(\Omega,\mathscr{F},P)$