lecture17.pdf

# lecture17.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 17 Review Class
Multiple Random Variables

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Multiple Random Variables Consider two random variables, X and Y mapping from Ω to R .
Multiple Random Variables Consider two random variables, X and Y mapping from Ω to R . 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 mapping from Ω to R . 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 ) Two discrete random variables X and Y are independent if and only if P ( X = a , Y = b ) = P ( X = a ) P ( Y = b ) for all a and b .
Conditioning Conditional PMF of X given Y : P ( X = i | Y = j ) = P ( { X = i }|{ Y = j } ) .

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Conditioning Conditional PMF of X given Y : P ( X = i | Y = j ) = P ( { X = i }|{ Y = j } ) . Compute P ( X | Y ) using the definition of conditional probability: P ( X = i | Y = j ) = P ( X = i , Y = j ) P ( Y = j ) since for any two events A , B we have P ( A | B ) = P ( A B ) P ( B ) .
Conditioning Conditional PMF of X given Y : P ( X = i | Y = j ) = P ( { X = i }|{ Y = j } ) . Compute P ( X | Y ) using the definition of conditional probability: P ( X = i | Y = j ) = P ( X = i , Y = j ) P ( Y = j ) since for any two events A , B we have P ( A | B ) = P ( A B ) P ( B ) . The conditional probability P ( X = i | Y = j ) is the joint probability P ( X = i , Y = j ) normalized by the marginal P ( Y = j ).

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Conditioning Conditional PMF of X given Y : P ( X = i | Y = j ) = P ( { X = i }|{ Y = j } ) . Compute P ( X | Y ) using the definition of conditional probability: P ( X = i | Y = j ) = P ( X = i , Y = j ) P ( Y = j ) since for any two events A , B we have P ( A | B ) = P ( A B ) P ( B ) . The conditional probability P ( X = i | Y = j ) is the joint probability P ( X = i , Y = j ) normalized by the marginal P ( Y = j ). An equivalent definition of independence is X and Y are independent if for all i , j , P ( X = i | Y = j ) = P ( X = i )
Suppose a student takes a Moodle quiz containing two questions. Let Q be the number of questions she gets correct. Let S be the number of hours she had studied that week. Suppose S takes the value 0, 1, or 2. If we pick a random student from the class, the joint probabilities of Q and S are as follows: S = 0 S = 1 S = 2 Q = 0 0.1 0.1 0 Q = 1 0.1 0.2 0 Q = 2 0 0 0.5 So, for example, P ( Q = 1 , S = 1) = 0 . 2. What will be P ( Q = 1) =?; P ( Q = 2) =?; P ( S = 1) =?; P ( S = 2) =?

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Suppose a student takes a Moodle quiz containing two questions. Let Q be the number of questions she gets correct. Let S be the number of hours she had studied that week. Suppose S takes the value 0, 1, or 2. If we pick a random student from the class, the joint probabilities of Q and S are as follows: S = 0 S = 1 S = 2 Q = 0 0.1 0.1 0 Q = 1 0.1 0.2 0 Q = 2 0 0 0.5 So, for example, P ( Q = 1 , S = 1) = 0 . 2. What will be P ( Q = 1) =?; P ( Q = 2) =?; P ( S = 1) =?; P ( S = 2) =? Are Q and S independent? Justify your answer.
Suppose a student takes a Moodle quiz containing two questions.

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