The Sample Space and Set Operations

The Sample Space

  • The sample space, \Omega, is the set of all possible outcomes, \omega.
  • Subsets of \Omega are events.
  • The empty set \emptyset contains no elements.

Example – Tossing a coin

Toss a coin once:
\Omega = \{H, T\}

Toss a coin twice:
\Omega = \{HH, HT, TH, TT\}

Then event that the first toss is heads: \omega = \{HH, HT\}

Set Operations – Complement, Union and Intersection

Complement

Given an event, A, the complement of A is A^\mathsf{c}, where:
A^\mathsf{c} = \text{"Not A"} = \{\omega \in \Omega : \omega \notin A\}

Union

The union of two sets A and B, A \cup B is set of the events which are in either A, or in B or in both.
A \cup B = \{\omega \in \Omega : \omega \in A \text{ or } \omega \in B \text{ or } \omega \in both\}
\bigcup_{i=1}^{\infty} A_i = \{\omega \in \Omega : \omega \in A_i \text{ for at least one i} \}

Intersection

The intersection of two sets A and B, A \cap B is set of the events which are in both A and B.
A \cap B = \{\omega \in \Omega : \omega \in A \text{ and } \omega \in B\}
\bigcap_{i=1}^{\infty} A_i = \{\omega \in \Omega : \omega \in A_i \text{ for all i} \}

Difference Set

The difference set is the events in one set which are not in the other:
A \setminus B = \{\omega : \omega \in A, \omega \notin B\}

Subsets

If every element of A is contained in B then A is a subset of B: A \subset B or equivalently, B \supset A.

Counting elements

If A is a finite set, then |A| denotes the number of elements in A.

Indicator function

An indicator function can be defined:
I_A(\omega) = I(\omega \in A) = \begin{cases}1\text{, }\omega \in A\\0\text{, otherwise}\end{cases}

Disjoint events

Two events A and B are disjoint or mutually exclusive if A \cap B = \emptyset (the empty set) – i.e. there are no events in both A and B).
More generally, A_1, A_2, ... are disjoint if A_i \cap A_j = \emptyset whenever i \neq j.

Example – intervals of the real line

The intervals A_1 = [0, 1), A_2 = [1, 2), A_3 = [2,3), ... are disjoint.
The intervals A_1 = [0, 1], A_2 = [1, 2], A_3 = [2, 3], ... are not disjoint. For example, A_1 \cap A_2 = \{1\}.

Partitions

A partition of the sample space \Omega is a set of disjoint events A_1, A_2, A_3, ... such that \bigcup_{i=1}^{\infty} A_i = \Omega.

Monotone increasing and monotone decreasing sequences

A sequence of events, A_1, A_2, ... is monotone increasing if A_1 \subset A_2 \subset A_3 \subset .... Here we define \lim_{n \to \infty} A_n = \bigcup_{i=1}^{\infty} A_i and write A_n \to A.

Similarly, a sequence of events, A_1, A_2, ... is monotone decreasing if A_1 \supset A_2 \supset A_3 \supset .... Here we define \lim_{n \to \infty} A_n = \bigcap_{i=1}^{\infty} A_i. Again we write A_n \to A

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