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Lectures 7 8 and 9

Lectures 7 8 and 9 - CHAPTER 3 Probabilities Associated...

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CHAPTER 3 Probabilities Associated with Random Variables We will revisit random variables again in the next chapter. But now it is time to complete the overall picture. We started with the notion of a random variable and its associated sample space. We then brought in the concept of an event, which is, simply, a subset of the sample space. Now we will complete the picture with an overview of probability. Once one has identified an event of interest, it is natural to ask: what is the probability that this event will occur? And so, in a way, the probability of an event is a kind of measure of the size of the event. But it is not size in the typically interpreted manner. We will begin this chapter with a discussion of what is meant by the term size , as related to the probability of an event. We will then proceed to more formally develop the notion of a probability measure associated with a random variable. . 3.1 Probability as a Measure of Size When most people think of the size of an object, they think of physical size. Size is an indicator of how small or large an object is. When the object is a subset of n -dimensional Euclidean space, n , then size is easily quantified; that is, it can be assigned a number. For example, consider an n -dimensional cube whose side has length, λ. It’s Euclidean size is then n λ . For n= 1, this is simply a line of length λ. For n =2 it is a square with area 2 . In 3-dimensional space it is a cube with volume 3 . For n >3 it can no longer be visualized, but we can extend the notion of size, and call it a hyper cube with “volume” n . Okay. So, what are some fundamental properties of this entity called size ? Well, we typically require size to be non-negative. Also, if a set is contained in another set, then the size of the former can be no bigger than the size of the latter. Still another reasonable property to require is that the size of two or more nonintersecting sets must equal the sum of the sizes of the individual sets. Now we are in a position to list the properties that the measure of size called probability must have. Definition 3.1 A measure of the size of events contained in the field of events, X , associated with a random variable, X , call it Pr(▪), is a probability measure if it satisfies all of the following properties:
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(P1) For any event, X A , 0 ) Pr( A . (P2)For any two disjoint, or non-intersecting sets ) Pr( ) Pr( ) Pr( , , B A B A B A X + = (P3) 1 ) Pr( 0 ) Pr( = = X S and . The first thing to note is that the operation Pr(▪) operates on the elements of the field of events, which are sets. And so, it is a measure of the “size” of a set. It does not measure the size of an element of the sample space, since the elements of the sample space are not sets. Finally, defining property (P3) simply requires that the “size” of the empty set be zero, and the “size” of the entire sample space be one; for, to allow the probability of an event to be greater than one (or 100%, if you like) is ridiculous.
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Lectures 7 8 and 9 - CHAPTER 3 Probabilities Associated...

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