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Unformatted text preview: Probabilities and Random Variables This is an elementary overview of the basic concepts of probability theory. I. The Probability Space The purpose of probability theory is to model random experiments so that we can draw inferences about them. The fundamental mathematical object is a triple ( , F , P ) called the probability space . A probability space is needed for each exper iment or collection of experiments that we wish to describe mathematically. The ingredients of a probability space are a sample space , a collection F of events , and a probability measure P . Let us examine each of these in turn. (a) The sample space . This is the set of all the possible outcomes of the experiment. Elements of are called sample points and typically denoted by . These examples should clarify its meaning: Example 1. If the experiment is a roll of a sixsided die, then the natural sample space is = { 1 , 2 , 3 , 4 , 5 , 6 } . Each sample point is a natural number between 1 and 6. Example 2. Suppose the experiment consists of tossing a coin three times. Let us write 0 for heads and 1 for tails. The sample space must contain all the possible outcomes of the 3 successive tosses, in other words, all triples of 0s and 1s: = { , 1 } 3 = { , 1 } { , 1 } { , 1 } = { ( x 1 , x 2 , x 3 ) : x i { , 1 } for i = 1 , 2 , 3 } = { (0 , , 0) , (0 , , 1) , (0 , 1 , 0) , (0 , 1 , 1) , (1 , , 0) , (1 , , 1) , (1 , 1 , 0) , (1 , 1 , 1) } . The four formulas are examples of different ways of writing down the set . A sample point = (0 , 1 , 0) means that the first and third tosses come out heads (0) and the second toss comes out tails (1). Example 3. Suppose the experiment consists of tossing a coin infinitely many times. Even though such an experiment cannot be physically arranged, it is of central 1 2 importance to the theory to be able to handle idealized situations of this kind. A sample point is now an infinite sequence = ( x 1 , x 2 , x 3 , . . . , x i , . . . ) = ( x i ) i =1 whose terms x i are each 0 or 1. The interpretation is that x i = 0 (1) if the i th toss came out heads (tails). In this model the infinite sequence of coin tosses is regarded as a single experiment, although we may choose to observe the individual tosses one at a time. (If you need a mental picture, think of the goddess of chance simultaneously tossing all infinitely many coins.) The sample space is the space of all infinite sequences of 0s and 1s: = { = ( x i ) i =1 : each x i is either 0 or 1 } = { , 1 } N . In the last formula N = { 1 , 2 , 3 , . . . } stands for the set of natural numbers, and the notation { , 1 } N stands for the infinite product set { , 1 } N = { , 1 } { , 1 } { , 1 } { , 1 } ....
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 Spring '11
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