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Unformatted text preview: Lecture 2. Constructing Probability Spaces This lecture describes some procedures for constructing probability spaces. We will work exclusively with discrete spacesusually finite ones. Later, we will see that the operations introduced here actually work for more general spaces. 2.1. Trees. Imagine the following experiment:- Pick a library at random.- Go there, and pick a shelf at random.- From that shelf, pick a book at random.- Open the book to a random page.- On that page, put your finger on a random word. We can visualize all the different ways in which you might complete this experiment by means of a tree diagram , a branching chart that illustrates the options. Draw a dot to represent a starting point. From this dot, draw several line segmentswe call them edges one for each of the libraries you might choose. Each edge ends at a dot labelled by a library. From each library dot, draw further edges corresponding to the shelves in that library; these terminate at dots labelled by the shelves. Each shelf dot has edges coming from it which end in dots labelled by the books on that shelf. (If there are several copies of a book on a given shelf, we will use just one edge for all the copies.) Each book dot has edges coming from it that end in page dots, and each page dot has edges from it that end in dots labelled by words. (If a word such as and appears several times on the page, we will use just one edge and terminal dot to represent it.) Each time you perform the experiment, you trace a path from the starting point, through a library, a shelf, a book and a page, finally ending on a word. Note that after splitting ways, paths never rejoin. The same word might occur at the end of many different paths, but the dots they are attached to are different. Suppose the choices at each possible stage have probabilities assigned to them. Per- haps there are three libraries: A , B and C , and the chances of choosing them are 1 / 2, 1 / 3 and 1 / 6, respectively. The set of all libraries, in other words, is a probability space. Again, each library has many shelves and each shelf has its own probability of being cho- sen randomly by someone in the library. In this way, the set of all shelves at library A becomes a probability space, and the same is true for the set of shelves in B and the set of shelves in C . Similarly, the books on each shelf have a probability space to represent their chances of being chosen. The same is true of the pages in each bookthey form a probability space. And finally, each page has an associated probability space. We may visualize this by labeling the edges emerging from any dot by the probabilities associated with the choices they represent....
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- Spring '08