Probability and Statistics

Sets and Events

In an experiment, the sample space is the set of all possible outcomes in which the subsets are considered events i.e. outcomes of interest.

Note

If $A\cap B$ = $\varnothing$, $A$ and $B$ are disjoint.

$\mathcal{F}$ is the collection of subsets of the events we’re interested in out of all possible outcomes in $\Omega$.

1. If $A,B\in \mathcal{F}$: $A\cup B\in \mathcal{F}$ and $A\cap B\in \mathcal{F}$
2. If $A\in \mathcal{F}$: $A^C\in \mathcal{F}$
3. $\varnothing \in \mathcal{F}$

Any $\mathcal{F}$ of subsets of $\Omega$ that satisfies the above is a field. Therefore, if $A_1,A_2,...,A_n\in \mathcal{F}$, then

$$\bigcup _{i=1}^n{A}_i\in F$$

i.e. $\mathcal{F}$ is closed under finite unions and finite intersections.

Note

I guess we want to assign a probability to all members of an event i.e. the subset?

Questions

• If $A\in \mathcal{F}$, how is the complement also in $\mathcal{F}$?
• Example 4 is confusing, not sure how the result is an infinite countable union?

Notation and Jargon