lec3 - p ( x ) p ( y ) p ( x y ) = p ( x ) p ( y | x ) = p...

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MIT OpenCourseWare http://ocw.mit.edu 8.044 Statistical Physics I Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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p x,y ( ζ , η ) ζ η p x ( ζ ) ζ 0 8.044 L3B1
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± p x,y ( ζ, η ) dζdη PROB( ζ< x ζ + and η< y η + ) P x,y ( ζ, η ) PROB( x ζ and y η ) ζ η = ± p x,y ( ζ ± ± ) ± ± −∞ −∞ p x,y ( ζ, η )= P x,y ( ζ, η ) ∂ζ ∂η 8.044 L3B2
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± ± ± ± f ( x, y ) ² = f ( ζ, η ) p x,y ( ζ, η ) dζdη −∞ −∞ NEW CONCEPTS: Reduction to a single variable p x ( ζ )= p x,y ( ζ, η ) −∞ Conditional probability density p x ( ζ y ) prob . ( ζ< x ζ + given that η = y ) | 8.044 L3B3
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( ) ( ) = ( ) 2 1 |y) 8.044 L3B4
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± 1 p x,y ( ζ, y ) = p y ( η = y ) ± ± −∞ 2 c p x ( ζ y ) = c ± ² −∞ ³´ | µ 1 c = p y ( η = y ) p x ( ζ | y )= p x,y ( ζ, y ) p ( x | y )= p ( x, y ) or p y ( η = y ) p ( y ) Bayes’ Theorem or “fundamental law of conditional probability” 8.044 L3B5
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p ( x, y )= p ( x | y ) p ( y ) p ( x, y )= p ( y | x ) p ( x ) x and y are statistically independent if p ( x, y )=
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Unformatted text preview: p ( x ) p ( y ) p ( x y ) = p ( x ) p ( y | x ) = p ( y ) | conditioned unconditioned 8.044 L3B6 F O a a y x a 2-y 2 = = O p ( x , y ) = 1 / a 2 p ( x , y ) = p ( y ) = 2 a ) 2 a 1 ( y 8 . 4 4 L 3 B 7 1 2 2 p ( x | y ) = 2 a 2 2 | x | a y y 2 = 0 x &gt; a 2 y | | p(x|y) a 2 -y 2 a 2 -y 2 x 8.044 L3B8 Derivation of Poisson density p(n;L) Given: p ( n = 1; x ) r x as x Start by nding p ( n = 0; L ) 0 L x d L dL 0 p (0) p (1) p ( n &gt; 1) p ( n = 0 , dL ) 1 p (1) = 1 r dL 8.044 L3B9 p ( n = 0; L ) = (1 r dL ) statistical independence ln p ( n = 0; L ) = ln(1 r dL ) L ( r dL ) ( r dL ) = rL 0 p ( n = 0; L ) = e rL p(n=0;L) 0 r-1 L 8.044 L3B10...
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lec3 - p ( x ) p ( y ) p ( x y ) = p ( x ) p ( y | x ) = p...

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