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ProbIntro1

# ProbIntro1 - This Bishopsslides Probability...

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his introduction is adapted from This introduction is adapted from Bishop’s slides

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robability Theory Probability Theory Apples and Oranges
robability Theory Probability Theory ( ) ,1 ij pxy = ∑∑ Marginal Probability Conditional Probability Joint Probability

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robability Theory Probability Theory Sum Rule Product Rule
e Rules of Probability The Rules of Probability Sum Rule Product Rule Independence ()( ) ( ) , pXY pXpY = () ( ) | pY X pY =

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ayes’ heorem Bayes Theorem posterior likelihood×prior
robability Densities Probability Densities

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ontinuous variables Continuous variables Y f XYdX = ( ) ( ) () , , | fY fX Y y fXY y f Y y = == = ( )
Transformed Densities and Expectation X EX μ == ⎣⎦ ( ) ( ) pxx d xp x x dx Conditional Expectation (discrete) Approximate Expectation (discrete and continuous) ( ) ( ) 2 22 2 var XX X XE X E X σμ μ ⎡⎤ = JFMS5

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| 2 2 | || ar | | o oY X x o YX EY X x x X x E Y X x μ μ = ⎡⎤ == = = ⎣⎦ = = ( ) var oo o x xo σμ ⎢⎥ var[ ] var | var | XX YE Y X E Y X =+ () | o pY X x = ( ) cov , XY ρ σ = 11 σσ ρ −≤ Observed correlation does NOT imply causation
rove the follwing () Prove the follwing var var var 2 cov , v 0 XY X Y X

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ProbIntro1 - This Bishopsslides Probability...

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