lecture18.pdf - COMPSCI 240 Reasoning Under Uncertainty...

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COMPSCI 240: Reasoning Under Uncertainty Arya Mazumdar University of Massachusetts at Amherst Fall 2016
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Lecture 18
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Probability Laws Probability law: assigns a probability P ( A ) to any event A Ω encoding our knowledge or beliefs about the collective “likelihood” of the elements of event A ; satisfies 3 axioms:
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Probability Laws Probability law: assigns a probability P ( A ) to any event A Ω encoding our knowledge or beliefs about the collective “likelihood” of the elements of event A ; satisfies 3 axioms: Nonnegativity: P ( A ) 0 for every A Ω
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Probability Laws Probability law: assigns a probability P ( A ) to any event A Ω encoding our knowledge or beliefs about the collective “likelihood” of the elements of event A ; satisfies 3 axioms: Nonnegativity: P ( A ) 0 for every A Ω Additivity: P ( A B ) = P ( A ) + P ( B ) if A and B are disjoint
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Probability Laws Probability law: assigns a probability P ( A ) to any event A Ω encoding our knowledge or beliefs about the collective “likelihood” of the elements of event A ; satisfies 3 axioms: Nonnegativity: P ( A ) 0 for every A Ω Additivity: P ( A B ) = P ( A ) + P ( B ) if A and B are disjoint Normalization: P (Ω) = 1
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Continuous Random Variables There are many random variables that are much more naturally thought of as taking continuous values than a finite or countable number of values (ex: height, weight, distance, time, speed, cost, etc...).
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Continuous Random Variables There are many random variables that are much more naturally thought of as taking continuous values than a finite or countable number of values (ex: height, weight, distance, time, speed, cost, etc...). Example: Suppose we measure the height of people to the nearest foot. This gives a discrete random variable with a probability mass function.
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Continuous Random Variables There are many random variables that are much more naturally thought of as taking continuous values than a finite or countable number of values (ex: height, weight, distance, time, speed, cost, etc...). Example: Suppose we measure the height of people to the nearest foot. This gives a discrete random variable with a probability mass function. Question: What happens to the probability mass function of height if we measure the people in smaller and smaller units?
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Example: Height - Nearest Foot
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Example: Height - Nearest 1/2 Foot
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Example: Height - Nearest 1/4 Foot
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Example: Height - Nearest 1/8 Foot
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Example: Height - Nearest 1/16 Foot
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Example: Height - Nearest 1/32 Foot
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Example: Height Question: What happens to the probability mass function if we measure the height of people in smaller and smaller units?
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Example: Height Question: What happens to the probability mass function if we measure the height of people in smaller and smaller units?
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  • Fall '12
  • Ben
  • Probability, Probability theory, probability density function, Subintervals

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