# lec3 - p x p y ⇒ p x y = p x p y | x = p y ± ²³ | ´...

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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
± 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
( ) ( ) = ( ) 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
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 ) ± ²³ | ´ ± ²³ ´ conditioned unconditioned 8.044 L3B6 F ± O ² a a y x a 2-y 2 = = O p ( x , y ) = 1 / π a 2 p ( x , y ) = p ( y ) = 2 a ) 2 π 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 > ± 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 ﬁnding p ( n = 0; L ) 0 L x d L dL → 0 ⇒ p (0) ± p (1) ± p ( n > 1) ⇒ p ( n = 0 , dL ) ≈ 1 − p (1) = 1 − r dL 8.044 L3B9 ± p ( n = 0; L ) = ² (1 − r dL ) statistical independence ln p ( n = 0; L ) = ³ ln(1 − r dL ) L → ³ ( − 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 y ± ²³ | ´...

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