DS_Lecture_30.pdf - Lecture 30 Bayes Rules Expected Value and Variance and Binormal Distribution Dr Chengjiang Long Computer Vision Researcher at

DS_Lecture_30.pdf - Lecture 30 Bayes Rules Expected Value...

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Lecture 30: Bayes Rules, Expected Value and Variance, and Binormal Distribution Dr. Chengjiang Long Computer Vision Researcher at Kitware Inc. Adjunct Professor at SUNY at Albany. Email: [email protected]
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C. Long Lecture 30 November 19, 2018 2 ICEN/ICSI210 Discrete Structures Outline Bayes Rule Expected Value and Variance Binominal Distribution
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C. Long Lecture 30 November 19, 2018 3 ICEN/ICSI210 Discrete Structures Outline Bayes Rule Expected Value and Variance Binominal Distribution
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C. Long Lecture 30 November 19, 2018 4 ICEN/ICSI210 Discrete Structures Law of Total Probability
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C. Long Lecture 30 November 19, 2018 5 ICEN/ICSI210 Discrete Structures Bayes Rule § x is the unknown cause § y is the observed evidence § Bayes rule shows how probability of x changes after we have observed y
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C. Long Lecture 30 November 19, 2018 6 ICEN/ICSI210 Discrete Structures Bayes Rule on the Fruit Example Suppose we have selected an orange. Which box did it come from?
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C. Long Lecture 30 November 19, 2018 7 ICEN/ICSI210 Discrete Structures Continuous Random Variables Examples: room temperature, time to run100m, weight of child at birth… Cannot talk about probability of that x has a particular value Instead, probability that x falls in an interval probability density function
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C. Long Lecture 30 November 19, 2018 8 ICEN/ICSI210 Discrete Structures Bayes Rule for Continuous Case
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C. Long Lecture 30 November 19, 2018 9 ICEN/ICSI210 Discrete Structures Outline Bayes Rule Expected Value and Variance Binominal Distribution
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C. Long Lecture 30 November 19, 2018 10 ICEN/ICSI210 Discrete Structures Expected Value and Variance All probability distributions are characterized by an expected value (mean) and a variance (standard deviation squared).
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C. Long Lecture 30 November 19, 2018 11 ICEN/ICSI210 Discrete Structures Expected value of a random variable Expected value is just the average or mean (µ) of random variable x . It’s sometimes called a “weighted average” because more frequent values of X are weighted more highly in the average. It’s also how we expect X to behave on-average over the long run (“frequentist” view again).
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C. Long Lecture 30 November 19, 2018 12 ICEN/ICSI210 Discrete Structures Expected value, formally å = x all ) ( ) p(x x X E i i Discrete case: Continuous case: dx ) p(x x X E i i ò = x all ) ( E(X) = µ. These symbols are used interchangeably
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C. Long Lecture 30 November 19, 2018 13 ICEN/ICSI210 Discrete Structures Example: expected value Recall the following probability distribution of ER arrivals: x 10 11 12 13 14 P(x) .4 .2 .2 .1 .1 å = = + + + + = 5 1 3 . 11 ) 1 (. 14 ) 1 (. 13 ) 2 (. 12 ) 2 (. 11 ) 4 (. 10 ) ( i i x p x
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C. Long Lecture 30 November 19, 2018 14 ICEN/ICSI210 Discrete Structures Sample Mean is a special case of Expected Value… Sample mean, for a sample of n subjects: ) 1 ( 1 1 n x n x X n i i n i i å å = = = = The probability (frequency) of each person in the sample is 1/n.
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C. Long Lecture 30 November 19, 2018 15 ICEN/ICSI210 Discrete Structures Example: the lottery A certain lottery works by picking 6 numbers from 1 to 49. It costs $1.00 to play the lottery, and if you win,
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