AMS 102.7 Spring 2006 Jingyu Zou Elements of Statistics Lecture Notes # 12

1 Sample space and events

Definition 1.1 Sample Space is the set of all possible outcomes of an experiment, denoted by Ω.

Example 1.2 If the experiment is simply tossing a coin, then Ω = {H,T } where H denotes head and T denotes tail.

Example 1.3 If we toss a die, then Ω = {1, 2, 3, 4, 5, 6}. Each number denotes a possible point from a single toss.

Example 1.4 Toss two coins. Ω = {HH,HT,TH,TT }. The outcome will be HH if both coins come up heads; it will be HT if the first coin comes up head and the second comes up tial; it will be TH if the first comes up tail and the second comes up head; and it will be TT if both comes up tails.

Definition 1.5 Any subset E of the sample space Ω is known as an event.

Example 1.6 In example 1.2, if E = {H}, then E is the event that a head appears on the flip of the coin. Similarly, if E = {T }, then E would be the event that a tail appears.

Example 1.7 In example 1.3, if E = {1}, then E is the event that one appears on the toss of the die. If E = {2, 4, 6}, then E would be the event that an even number appears on the toss.

Example 1.8 In example 1.4, if E = {HH,HT }, then E is the event that a head appears on the first coin.

For any two events E and F of a sample space Ω, E ∪ F is the event consisting of all points which are either in E or in F or in both E and F . That is, the event E ∪ F will occur if either E or F occurs. For any two events E and F , E ∩ F is the event consisting of all points which are both in E and F . That is, the event E ∩ F will occur only if both E and F occurs. E ∩ F can be simply written as EF . If EF = ∅, then E and F are said to be mutually exclusive. For any event E, EC is the event consisting of all points in the sample space Ω which are not in E. That is EC will occur if and only if E does not occur.

1 2 Probability

Consider a sample space Ω. For each event E of the sample space Ω, we assume that a number P (E) is defined as satisfies the following three conditions: (1) 0 ≤ P (E) ≤ 1 for any event E (2) P (Ω) = 1 (3) For any sequence of events E1,E2,...,En which are mutually exclusive, that is, events for which Ei ∩ Ej = ∅ when i 6= j, then

P (E1 ∪ E2 ∪ ... ∪ En) = P (E1) + P (E2) + ... + P (En)

Corollary 2.1 (1) The probability of empty event is 0. That is, P (∅) = 0 (2) If event E is a subset of event F , that is, E ⊂ F , then P (E) ≤ P (F )

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