49 example what is the analog for uncountable sample

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EXAMPLE : What is the analog for uncountable sample spaces of the equally likely case for discrete sample spaces? Suppose   0, c for some c 0. Write P a b for the probability of event : a b . Then assume the probability is P a b b a c for any 0 a b c . This is the length of the interval a , b over the length of the entire interval for the sample space. 50
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In this example, all particular values have zero probability of occurring, that is, P  a  0 for all possible outcomes 0 a c . Why? Suppose a is strictly between 0 and c ; the endpoints are handled similarly. Then for integers j large enough so that a 1/ j 0 and a 1/ j c , P a 1/ j a 1/ j 2/ cj . But P  a  P a 1/ j a 1/ j because a a 1/ j , a 1/ j . So for all j sufficiently large, P  a  2/ cj , which can only happen if P  a  0. 51
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Uncountable spaces are intuitively tricky: any particular outcome has zero probability (yet, when we turn to statistics and collecting data, we eventually see an outcome). In such cases, we focus on the probability of events such as intervals. For the “equally likely” model, the probability of events involving intervals of the same length is the same – regardless of where the interval starts. For example, if c 100, P 0 20 P 80 100 20/100 1/5. Later, we use the notion of continuous random variables to represent uncountable sample spaces. 52
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