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lecture18.pdf

# 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
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
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
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.
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
Example: Height - Nearest 1/2 Foot

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Example: Height - Nearest 1/4 Foot
Example: Height - Nearest 1/8 Foot

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Example: Height - Nearest 1/16 Foot
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?
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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