lecture10.pdf

lecture10.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 10
How difficult was the midterm As of 2:20 pm: 91 voted

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How difficult was the midterm As of 2:20 pm: 91 voted Extremely difficult 44
How difficult was the midterm As of 2:20 pm: 91 voted Extremely difficult 44 Difficult but expected 36

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How difficult was the midterm As of 2:20 pm: 91 voted Extremely difficult 44 Difficult but expected 36 As expected 10
How difficult was the midterm As of 2:20 pm: 91 voted Extremely difficult 44 Difficult but expected 36 As expected 10 Easier than expected 1

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How difficult was the midterm As of 2:20 pm: 91 voted Extremely difficult 44 Difficult but expected 36 As expected 10 Easier than expected 1 Super easy 1
Recall: Random Variable A random variable is a function that maps from the sample space to the real numbers, X : Ω R

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Multiple Random Variables
Multiple Random Variables Consider two random variables, X and Y mapping from Ω to R .

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Multiple Random Variables Consider two random variables, X and Y mapping from Ω to R . X defines events of the form { X = i } = { o Ω | X ( o ) = i } and Y defines events of the form { Y = j } = { o Ω | Y ( o ) = j }
Multiple Random Variables Consider two random variables, X and Y mapping from Ω to R . X defines events of the form { X = i } = { o Ω | X ( o ) = i } and Y defines events of the form { Y = j } = { o Ω | Y ( o ) = j } For i , j R , we can define the event { X = i , Y = j } = { X = i } ∩ { Y = j }

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Multiple Random Variables Consider two random variables, X and Y mapping from Ω to R . X defines events of the form { X = i } = { o Ω | X ( o ) = i } and Y defines events of the form { Y = j } = { o Ω | Y ( o ) = j } For i , j R , we can define the event { X = i , Y = j } = { X = i } ∩ { Y = j } = { o Ω | X ( o ) = i and Y ( o ) = j }
Multiple Random Variables Consider two random variables, X and Y mapping from Ω to R . X defines events of the form { X = i } = { o Ω | X ( o ) = i } and Y defines events of the form { Y = j } = { o Ω | Y ( o ) = j } For i , j R , we can define the event { X = i , Y = j } = { X = i } ∩ { Y = j } = { o Ω | X ( o ) = i and Y ( o ) = j } The probabilities of these events give the joint PMF of X and Y : P ( X = i , Y = j ) = P ( { X = i , Y = j } ) = p X , Y ( i , j )

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Multiple Random Variables Consider two random variables, X and Y
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• Fall '12
• Ben
• Variance, Probability theory, probability density function

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