November 2012
Intermediate to advanced
794 pages
22h 9m
English
book
Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications
by John J. Shynk
Content preview from Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications
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B.2 SUPREMUM AND INFIMUM
Definition: Supremum The supremum of set 
is the least upper bound: it is the smallest real number that is ≥ all elements in E.
The supremum is useful when working with intervals because
(B.21) 
The maximum is not defined for E = [a, b) because it must be an element of the set; b is not in this set, and for any 
for 
, we can always find an element of E that is greater. The maximum is also called the greatest element. If a set has a greatest element, then the supremum and maximum are identical.
Definition: Infimum The infimum of set 
is the greatest lower bound: it is the largest real number that is ≤ all elements in E.
Thus
(B.22) 
The minimum of a set is called the least element and is not defined for semi-open intervals such as (a, b]. If a set has a least element, then the infimum and minimum are identical.
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