L18_VarianceRVs_print

L18_VarianceRVs_print - 1 COMP170 Discrete Mathematical...

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Unformatted text preview: 1 COMP170 Discrete Mathematical Tools for Computer Science Discrete Math for Computer Science K. Bogart, C. Stein and R.L. Drysdale Section 5.7, pp. 294-303 Variance of RVs Version 2.0: Last updated, May 13, 2007 Slides c 2005 by M. J. Golin and G. Trippen 2 Probability Distributions and Variance • Distributions of Random Variables • Variance 3 Distributions of Random Variables Expected value Example: Flip a coin 100 times, expected number of H is 50 . To what extent do we expect to see 50 heads? Is it surprising to see 55 , 60 , or 65 heads instead? General Question: how much do we expect a random variable to deviate from its expected value. 4 The distribution function D of a random variable X with finitely many values is the function on the values of X defined by D ( x ) = P ( X = x ) . The distribution function of the random variable X assigns to each value x of the random variable the probability that X achieves that value. Visualize the distribution function using a diagram called a histogram . Graphs that show, for each integer value x of X , a rectangle of width 1 centered at x , whose height (and thus area) is proportional to the probability P ( X = x ) . 5 Examples: 10 coin flips Ten-question test with probability . 8 of getting a correct answer. Area of Rectangles with bases ranging from x = a to x = b is probability that X is between a and b . Cumulative distribution function D : D ( a, b ) = P ( a ≤ X ≤ b ) . 6 With more coin flips or more questions, will the results spread out? 25 trials: Expected number of heads is 12 . 5 . Histogram says: vast majority of results between 9 and 16 heads. Virtually all results lie between 5 and 20 . Thus, results are not spread as broadly (relatively speaking) as they were with just 10 flips. 7 Test score histogram with 25 questions Compared to coin flipping, even more tightly packed around its expected value. Essentially, all scores lie between 14 and 25 . 8 100 flips of a coin 100-question test Two histograms have almost same shape. Number of heads has virtually no chance of deviating by more than 15 from its expected value, and test score has almost no chance of deviating by more than 11 . Thus, spread has only doubled , even though number of trials has quadrupled . 9 We need about 30 values to see the most relevant probabilities for 100 trials, whereas we need 15 values to see the most relevant probabilities for 25 independent trials. How many values to see essentially all results in 400 trials. Only about 60 values. 10 In both cases, curve formed by tops of rectangles seems quite similar to bell-shaped curve, called normal curve . We want an algebraic way to measure the difference between a random variable and its expected value....
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This note was uploaded on 08/25/2010 for the course COMP COMP170 taught by Professor M.j.golin during the Spring '10 term at HKUST.

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L18_VarianceRVs_print - 1 COMP170 Discrete Mathematical...

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