Essay About Set P And Toss Of A Coin

Introduction to Financial Mathematics
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Introduction to Financial Mathematics
Lecture Notes — MAP 5601
Department of Mathematics
Florida State University
Fall 2003
Table of Contents
Lecture Notes — MAP 5601 map5601LecNotes.tex i 8/27/2003
1. Finite Probability Spaces
The toss of a coin or the roll of a die results in a finite number of possible outcomes.
We represent these outcomes by a set of outcomes called a sample space. For a coin we
might denote this sample space by {H, T} and for the die {1, 2, 3, 4, 5, 6}. More generally
any convenient symbols may be used to represent outcomes. Along with the sample space
we also specify a probability function, or measure, of the likelihood of each outcome. If
the coin is a fair coin, then heads and tails are equally likely. If we denote the probability
measure by P, then we write P(H) = P(T) = 1
2 . Similarly, if each face of the die is equally
Defninition 1.1. A finite probability space is a pair (
, P) where
is the sample space set
and P is a probability measure:
= {!1, !2, . . . , !n}, then
(i) 0 < P(!i) 1 for all i = 1, . . . , n (ii) n Pi=1 P(!i) = 1. In general, given a set of A, we denote the power set of A by P(A). By definition this Here, as always, ; is the empty set. By additivity, a probability measure on extends to ) if we set P(;) = 0. 6 , while P (toss is even) = P (toss is odd) = 1 6 = 1 a partition. Defninition 1.2. A partition of a set (of arbitrary cardinality) is a collection of nonempty disjoint subsets of whose union is If the outcome of a die toss is even, then it is an element of {2, 4, 6}. In this way partitions may provide information about outcomes. Defninition 1.3. Let A be a partition of . A partition B of is a refinement of A if every member of B is a subset of a member of A. Lecture Notes -- MAP 5601 map5601LecNotes.tex 1 8/27/2003 that a refinement contains at least as much information as the original partition. In the language of probability theory, a function on the sample space is called a random variable. This is because the value of such a function depends on the random occurrence of a point of . However, without this interpretation, a random variable is just a function. Given a finite probability space ( , P) and the real-valued random variable X : ! IR we define the expected value of X or expectation of X to be the weighted probability of its values. Definition 1.4. The expectation, E(X), of the random variable X : ! IR is by definition E(X) = n Xi=1 X(!i)P(!i) where = {!1, !2, . . . , !n}. We see in this definition an immediate utility of the property n Pi=1 P(!i) = 1. If X is identically constant, say X = C, then E(X) = C. When a partition of is given, giving more informaiton in general than just , we define a conditional expectation. Definition 1.5. Given a finite probability space ( , P) and a partition of , A,