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concepts1 - Discrete Random Variables October 15, 2009 Let...

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Unformatted text preview: Discrete Random Variables October 15, 2009 Let S be a sample space. Recall that if A S is an event, P ( A ) denotes the probability of A . (Probability is a function on events .) 0.1 Random Varaibles, Probablity Mass Function, Expec- tation and Variance. In this discussion, we will assume that S is discrete (i.e., finite or countable). We shall see later (i.e., in Chapter 6) that this assumption is not essentialthat is to say, we can develop a perfectly sound theory of random variables without it. However, the assumption is often satisfied in practice. When this is true, it is possible to compute the probabilities of all events from the probabilities of the outcomes they contain (Fact 1, below). This simplifes some proofs (e.g., Fact 3, below.) Fact 1. If A is an event, P ( A ) = s A P ( { s } ) Definition. A function from S to R is called a random variable . Remark. We really should have mentioned the requirement that X be measur- able . We will see the need for this assumption later. For the time being, it is not essential to include this stipulation, since all real-valued functions on a discrete set are measurable. Remark. Note that given any sample space (discrete or not), and any random variable, we may make a coarser sample space from S by treating the events of the form { s S | X ( s ) = x } , x R , as outcomes. As long as we ask no questions that concern events that are any finer (i.e., smaller) than these, we lose nothing. Remark. We say that X is discrete if its set of values { X ( s ) | s S } is discrete. Certainly, if S is discrete, then so is X . In case only X is discrete, then by the maneuver in the last remark, we may replace the sample space on which X is defined with a discrete one. This shows that confining attention to discrete sample spaces in not a serious limitation....
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This note was uploaded on 11/29/2011 for the course MATH 3355 taught by Professor Britt during the Spring '08 term at LSU.

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concepts1 - Discrete Random Variables October 15, 2009 Let...

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