Measurable Sets
At any given time point, we are interested in knowing what the future will be at a later time point. But the future is full of uncertainties: tomorrow it may rain or may not rain; the lottery ticket you are buying now may win or may not win. Just like flipping a coin, we will not know which scenario would eventually become true, until the coin has been flipped. Thus, in the presence of uncertainties, instead of asking what the future will be, we shall ask what are the possible scenarios and what are their chances to become true in the future. This leads us to the realm of Probability Theory. Modern Probability Theory is built over a triple (Ω, F, P), where Ω is the collection of all possible scenarios for the future, F collects all the events that are of interest, and P tells the probability of each event in F. We start with exploring F in this chapter.
1. Definition and basic properties
Suppose that we are going to conduct an experiment of flipping a coin three times. We use H and T to denote head and tail, respectively. Then we are facing eight possible outcomes in the future: HHH, HHT, HTH,HTT, THH,THT,TTH,TTT. We collect them together and denote it by a set
(1.1) Ω := HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }.
Consider the event that the first flip is H. It means precisely that if we have conducted the experiment, then our final outcome would be one of the following four: HHH, HHT, HTH,HTT. We may thus use the set
FH := HHH, HHT, HTH, HTT
to denote the event that the first flip is H. Other events can be similarly identified as subsets of Ω as well. For example, we identify the event that the second flip is H with the set SH := {HHT, HHH, THH, THT }.
Given that we use subsets of Ω to denote events, how do we formulate
the occurrence of an event mathematically? Say, we have conducted the experiment of flipping the coin three times, and the final outcome is ω
(which of course is an element in Ω). The event that the first flip is H has occurred means precisely that the realized outcome ω is one of the four:
HHH, HHT, HTH,HTT. Thus the event FH occurs at a realization ω iff ω ∈ FH.
Let’s now consider the collection of all interesting events. Say, suppose that, for some reasons, we care about only the first two flips. Take any event E in the collection of interesting events. At an arbitrary realization ω, E occurs iff ω ∈ E iff ω /∈ Ec iff Ec does not occur. That is, Ec is the contrary of E. Intuitively, we shall care about the contrary of an interesting event. Thus Ec should also be included in our collection of interesting events. For example, if E = SH, then its complement ST := {HTH, HTT, TTH, TTT } is the event that the second flip is T, which is of course interesting. We may also look at multiple interesting events together. Let’s take two interesting events E and F . At an arbitrary realization ω, the intersection E ∩ F occurs iff ω ∈ E ∩ F iff ω ∈ E and ω ∈ F iff both E and F occur. Intuitively, we shall care about the simultaneous occurrence of two interesting events. Thus E ∩ F should lie in our collection of interesting events as well. For example, for the event FH that the first flip is T and the event ST that the second flip is T, the intersection FH ∩ ST = {HTH, HTT } is precisely the event that the first flip is H and the second flip is T, which is again obviously interesting to us as we care about the first two flips.
The above discussions motivate us to conclude that our collection of interesting events should be closed under taking complementation and in- tersection. This leads us to the following notion.
1.1. DEFINITION. Let Ω be a non-empty set. Let F be a collection of
subsets of Ω. We say that F is a σ-algebra over Ω if it satisfies the following
conditions:
CS 340 Milestone One Guidelines and Rubric Overview: For this assignment, you will implement the fundamental operations of create, read, update,
Retail Transaction Programming Project Project Requirements: Develop a program to emulate a purchase transaction at a retail store. This
7COM1028 Secure Systems Programming Referral Coursework: Secure
Create a GUI program that:Accepts the following from a user:Item NameItem QuantityItem PriceAllows the user to create a file to store the sales receip
CS 340 Final Project Guidelines and Rubric Overview The final project will encompass developing a web service using a software stack and impleme
