Statistical Inference/Probability Theory: Difference between revisions
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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). | ||
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. | |||
We first need to define formally the following two relationships, which allow us to order and equate sets: | |||
=== Complementation = | '''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> | |||
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>A^c</amsmath>, is the set of all elements that are not in A: | |||
<amsmath>A^c = \{ x : x \notin A \}</amsmath> | |||
== Event Operations [Theorem 1.1.4] == | == Event Operations [Theorem 1.1.4] == |
Revision as of 10:49, 1 August 2018
Set Theory
Sample Space [Definition 1.1.1]
The set, S, of all possible outcomes of a particular experiment is called the sample space for the experiment.
Event [Definition 1.1.2]
An event is any collection of possible outcomes of an experiment, that is, any subset of S (including S itself).
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.
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>
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>A^c</amsmath>, is the set of all elements that are not in A:
<amsmath>A^c = \{ x : x \notin A \}</amsmath>