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Slides03-2010

Slides03-2010 - Probability Theory and Measure Lecture III...

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Probability Theory and Measure: Lecture III Charles B. Moss August 26, 2010 Charles B. Moss () Probability Theory and Measure: Lecture III August 26, 2010 1 / 13

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Outline 1 Uniform Probability Measure Lebesgue Measure 2 Lebesgue Measure and Lebesgue Integral 3 Random Variables and Distributions Charles B. Moss () Probability Theory and Measure: Lecture III August 26, 2010 2 / 13
Uniform Probability Measure Uniform Probability Measure I think that Bierens discussion of the uniform probability measure provides a firm basis for the concept of probability measure. First, we follow the conceptual discussion of placing ten balls numbered 0 through 9 into a container. Next, we draw out an infinite sequence of balls out of the container, replacing the ball each time. In Excel, we can mimic this sequence using the function floor(rand()*10,1). This process will give a sequence of random numbers such as presented in Table 1. Charles B. Moss () Probability Theory and Measure: Lecture III August 26, 2010 3 / 13

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Uniform Probability Measure Table: Random Draws of Single Digits Ball Drawn Draw 1 Draw 2 Draw 3 1 7 0 3 2 4 2 0 3 1 9 2 4 4 6 2 5 8 4 0 6 3 5 4 Charles B. Moss () Probability Theory and Measure: Lecture III August 26, 2010 4 / 13
Uniform Probability Measure Taking each column, we can generate three random numbers 0.741483, 0.029645, 0.302204. Note that each of these sequences are contained in the unit interval Ω = [0 , 1]. The primary point of the demonstration is that the number draw { x Ω = [0 , 1] } is a probability measure. Taking x = 0 . 741483 as the example, we want to prove that P ([0 , x = 0 . 741483]) = 0 . 741483. To do this we want to work out the probability of drawing a number less than 0.741483.

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Slides03-2010 - Probability Theory and Measure Lecture III...

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