Sample Space

Definition 1.1 (Random Experiment)

(a) All outcomes of the experiment should be known in advance

(b) any performance of the experiment results in an outcome that is not known in advance, and

(c) the experiment can be repeated under identical conditions

Definition 1.2 (Sample Space)

sample space of a statistical experiment is a pair $(\Omega,S)$, where

(a) $\Omega$ is the set of all possible outcomes of the experiment and

(b) $S$ is a $\sigma$-field of subsets of $\Omega$

Definition 1.3 (Sample Points and Events)

The elements of $\Omega$ are called sample points. Any set $A\in S$ is called an event

Definition 1.4 (Probability axioms)

Let $(\Omega,S)$ be the sample space associated with a statistical experiment. A set function $P$ defined on $S$ is called a probability measure if it satisfied the following conditions:

1. $P(A)\geq0$ for all $A\in S$

2. $P(\Omega)=1$

3. Given disjoint sets, $P(\sum\limits^{\infty}{j=1}A_j)=\sum\limits^{\infty}{j=1}P(A_j)$

Remark 1.5 (Probability of event)

$P(A)$ is the probability of event $A$

Definition 1.6 (Probability space)

triple $(\Omega,S,P)$ is called a probability space

Definition 1.7 (Sigma algebra)

A $\sigma$-algebra $F$ of subsets of X is a collection $F$ of subsets of $X$ satisfying the following conditions

1. $\empty\sub F$

2. If $B\sub F$, $B^C\sub F$

3. For a countable collection of sets in $F$ , its union is is $F$

Example 1.8 (Example of sigma algebra of subsets of $X$)

1. $\{\empty,X\}$ is sigma-algebra of $X$

2. $P(X)$ is sigma-algebra of $X$

3. For , Sigma-algebra of $X$, $\{\empty,X\}\sub \mathcal{F} \sub P(X)$

Proposition 1.9 (Countable Intersection in sigma algebra)

Let $\mathcal{F}$ be a sigma algebra of subsets of $X$.

Then, If $A_1,A_2,...$ are countable collection of sets in $\mathcal{F}$ , then $\cap^{\infty} A_n \sub \mathcal{F}$

Proof )

$A_1^C,A_2^C,... \sub \mathcal{F}$

Therefore, countable union of the complements of $A_i^C$ are in $\mathcal{F}$ and its complement is $\cap^{\infty}A_n$ is in

$\mathcal{F}$

Definition 1.10 (Generated Sigma Algebra)

Let $X$ be a set and $\mathcal{B}$ a non empty subset of $X$. The smallest sigma algebra containing all set sets of $\mathcal{B}$ is called sigmal algebra generated by the collection $\mathcal{B}$ and is denoted $\sigma (\mathcal{B})$

Definition 1.11 (Measurable Space)

Let $\Omega$ set of outcomes and $\mathcal{F}$ a sigma algebra of subsets of $\Omega$. Then $(\Omega,\mathcal{F})$ is a measurable space

Definition 1.12 (Measure)

A measure is a nonnegative countable additive set function.

$\mu:\mathcal{F}\rightarrow R$ is a function which satisfies,m

$1) \mu(A)\geq\mu(\empty)=0$

1. For a countable sequence of disjoint set in $\mathcal{F}$, $\mu(\cup A_i)=\sum\mu(A_i)$

If $\mu(\Omega)=1$, Then , $\mu$ is a probability measure.

Definition 1.13 (Borel Set)

(a)Borel Sigma Algebra is sigma-algebra generated by the collection of open sets

(b) Consider $\Omega=(0,1]$. Let $C_0$ be the collection of all open intervals in $(0,1]$. Then $\sigma (C_0)$ , the sigma algebra generated by $C_0$ is called the Borel sigma algebra. It is denoted by $\mathcal{B}((0,1])$

(b) An element of $\mathcal{B}((0,1])$ is called a Borel measurable set, or simply a Borel set

Lemma 1.14 (Singleton is Borel Set)

By using countable union and complement property of sigma-algebra, we can show that singleton is Borel Set. Use $(b-\frac{1}{n},b+\frac{1}{n})$. We can also show that closed interval is Borel Set.

reference

an introduction to probability and statistics ( rohatgi )