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L18_VarianceRVs - COMP170 Discrete Mathematical Tools for...

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1-1 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
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2-1 Probability Distributions and Variance
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2-2 Probability Distributions and Variance Distributions of Random Variables
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2-3 Probability Distributions and Variance Distributions of Random Variables Variance
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3-1 Distributions of Random Variables
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3-2 Distributions of Random Variables Expected value
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3-3 Distributions of Random Variables Expected value Example: Flip a coin 100 times, expected number of H is 50 .
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3-4 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?
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3-5 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?
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3-6 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.
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4-1 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 ) .
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4-2 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.
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4-3 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 .
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4-4 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 ) .
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5-1 Examples:
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5-2 Examples: 10 coin flips
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5-3 Examples: 10 coin flips
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5-4 Examples: 10 coin flips Ten-question test with probability . 8 of getting a correct answer.
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5-5 Examples: 10 coin flips Ten-question test with probability . 8 of getting a correct answer.
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5-6 Examples: 10 coin flips Ten-question test with probability . 8 of getting a correct answer.
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