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lec-30 - Administration 1 Midterm 2 Wednesday November 9th...

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Administration 1. Midterm 2, Wednesday, November 9th. 8-10PM, 155 Dwinelle. 2. Extra office hours: Tuesday, November 8th, 4-9PM, 310 Soda. 3. No class on Wednesday!!
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CS70: Satish Rao: Lecture 30. 1. Review Joint distribution, conditional expectation. 2. Conditional Independence. 3. Inference.
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Joint Distributions. For random variables, X and Y the joint distribution is Pr [ X = a , Y = b ] for each possible a , b .
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Joint Distributions. For random variables, X and Y the joint distribution is Pr [ X = a , Y = b ] for each possible a , b . Marginal Distributions are the distibutions of X and Y . ( Pr [ X = a ] = b Pr [ X = a , Y = b ] . )
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Joint Distributions. For random variables, X and Y the joint distribution is Pr [ X = a , Y = b ] for each possible a , b . Marginal Distributions are the distibutions of X and Y . ( Pr [ X = a ] = b Pr [ X = a , Y = b ] . ) Conditional Expectation of X given event A ] E ( X | A ) = a a × Pr [ X = a | A ] .
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Joint Distributions. For random variables, X and Y the joint distribution is Pr [ X = a , Y = b ] for each possible a , b . Marginal Distributions are the distibutions of X and Y . ( Pr [ X = a ] = b Pr [ X = a , Y = b ] . ) Conditional Expectation of X given event A ] E ( X | A ) = a a × Pr [ X = a | A ] . Total Expectation Law: E ( X ) = b Pr [ Y = b ] E ( X | Y = b ) .
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Conditional independence. Events A and B are conditionally independent given C if Pr [ A , B | C ] = Pr [ A | C ] × Pr [ B | C ] . or Pr [ A | B , C ] = Pr [ A | C ] . Example: roll 3 die.
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Conditional independence. Events A and B are conditionally independent given C if Pr [ A , B | C ] = Pr [ A | C ] × Pr [ B | C ] . or Pr [ A | B , C ] = Pr [ A | C ] . Example: roll 3 die. A - “sum of first two greater than 7”
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Conditional independence. Events A and B are conditionally independent given C if Pr [ A , B | C ] = Pr [ A | C ] × Pr [ B | C ] . or Pr [ A | B , C ] = Pr [ A | C ] . Example: roll 3 die. A - “sum of first two greater than 7” B - “sum of first and third is greater than 7”
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Conditional independence. Events A and B are conditionally independent given C if Pr [ A , B | C ] = Pr [ A | C ] × Pr [ B | C ] . or Pr [ A | B , C ] = Pr [ A | C ] . Example: roll 3 die. A - “sum of first two greater than 7” B - “sum of first and third is greater than 7” C - “first one is 5” A and B are not independent.
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Conditional independence. Events A and B are conditionally independent given C if Pr [ A , B | C ] = Pr [ A | C ] × Pr [ B | C ] . or Pr [ A | B , C ] = Pr [ A | C ] . Example: roll 3 die. A - “sum of first two greater than 7” B - “sum of first and third is greater than 7” C - “first one is 5” A and B are not independent. Pr [ A | B ] > Pr [ A ]
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Conditional independence. Events A and B are conditionally independent given C if Pr [ A , B | C ] = Pr [ A | C ] × Pr [ B | C ] . or Pr [ A | B , C ] = Pr [ A | C ] . Example: roll 3 die. A - “sum of first two greater than 7” B - “sum of first and third is greater than 7” C - “first one is 5” A and B are not independent. Pr [ A | B ] > Pr [ A ] Pr [ A , B | C ] = Pr [ A | B ] × Pr [ B | C ]
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Conditional independence. Events A and B are conditionally independent given C if Pr [ A , B | C ] = Pr [ A | C ] × Pr [ B | C ] . or Pr [ A | B , C ] = Pr [ A | C ] .
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