Lecture 03-2007 - Probability Theory and Measure: Lecture...

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1 Probability Theory and Measure: Lecture III I. Uniform Probability Measure: A. I think that Bieren’s discussion of the uniform probability measure provides a firm basis for the concept of probability measure. 1. 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. 2. 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: Table 1. 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 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   0,1  . The primary point of the demonstration is that the number draw (   0,1 x   ) is a probability measure. a. Taking 0.741483 x as the example, we want to prove that     0, 0.741483 0.741483 Px  . To do this we want to work out the probability of drawing a number less than 0.741483. b. As a starting point, what is the probability of drawing the first number in Table 1 less than 7, it is 7 ~{0,1,2,3,4,5,6}. Thus, without consider the second number, the probability of drawing a number less than 0.741483 is somewhat greater than 7/10. c. Next, we consider drawing a second number given that the first number drawn is greater than or equal to 7. Now, we are interested
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AEB 6933 Mathematical Statistics for Food and Resource Economics Lecture III Professor Charles B. Moss
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Lecture 03-2007 - Probability Theory and Measure: Lecture...

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