Statistical Inference/Probability Theory: Difference between revisions
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The set, ''S'', of all possible outcomes of a particular experiment is called the ''sample space'' for the experiment. [Definition 1.1.1] | The set, ''S'', of all possible outcomes of a particular experiment is called the ''sample space'' for the experiment. [Definition 1.1.1] | ||
== Event | == Event == | ||
An ''event'' is any collection of possible outcomes of an experiment, that is, any subset of ''S'' (including ''S'' itself). | An ''event'' is any collection of possible outcomes of an experiment, that is, any subset of ''S'' (including ''S'' itself). [Definition 1.1.2] | ||
Let ''A'' be an event, a subset of ''S''. We say the event ''A'' occurs if the outcome of the experiment is in the set ''A''. When speaking of probabilities, we generally speak of the probability of an event, rather than a set. But we may use the terms interchangeably. | Let ''A'' be an event, a subset of ''S''. We say the event ''A'' occurs if the outcome of the experiment is in the set ''A''. When speaking of probabilities, we generally speak of the probability of an event, rather than a set. But we may use the terms interchangeably. |
Revision as of 13:02, 1 August 2018
Set Theory
Sample Space
The set, S, of all possible outcomes of a particular experiment is called the sample space for the experiment. [Definition 1.1.1]
Event
An event is any collection of possible outcomes of an experiment, that is, any subset of S (including S itself). [Definition 1.1.2]
Let A be an event, a subset of S. We say the event A occurs if the outcome of the experiment is in the set A. When speaking of probabilities, we generally speak of the probability of an event, rather than a set. But we may use the terms interchangeably.
Base relationships
We first need to define formally the following two relationships, which allow us to order and equate sets:
Containment:
<amsmath>A \subset B \Leftrightarrow x \in A \Rightarrow x \in B</amsmath>
Equality:
<amsmath>A = B \Leftrightarrow A \subset B \land B \subset A</amsmath>
Elementary operations
Given any two events (or sets) A and B , we have the following elementary set operations:
Union: The union of A and B, written <amsmath>A \cup B</amsmath>, is the set of elements that belong to either A or B or both:
<amsmath>A \cup B = \{ x : x \in A \lor x \in B \}</amsmath>
Intersection: The intersection of A and B, written <amsmath>A \cap B</amsmath>, is the set of elements that belong to both A and B:
<amsmath>A \cap B = \{ x : x \in A \land x \in B \}</amsmath>
Complementation: The complement of A, written <amsmath>\overline{A}</amsmath>, is the set of all elements that are not in A:
<amsmath>\overline{A} = \{ x : x \notin A \}</amsmath>
Event Operations
The elementary set operations can be combined: for any three events, A, B, and C, defined on a sample space S, the following relationships hold [Theorem 1.1.4].
Commutativity
- <amsmath>A \cup B = B \cup A</amsmath>
- <amsmath>A \cap B = B \cap A</amsmath>
Associativity
- <amsmath>A \cup (B \cup C) = (A \cup B) \cup C</amsmath>
- <amsmath>A \cap (B \cap C) = (A \cap B) \cap C</amsmath>
Distributive Laws
- <amsmath>A \cap (B \cup C) = (A \cap B) \cup (A \cap C)</amsmath>
- <amsmath>A \cup (B \cap C) = (A \cup B) \cap (A \cup C)</amsmath>
DeMorgan Laws
- <amsmath>\overline{A \cup B} = \overline{A} \cap \overline{B}</amsmath>
- <amsmath>\overline{A \cap B} = \overline{A} \cup \overline{B}</amsmath>