lec5 - T p( ) ( / ) ( / ) 2 e ( / ) 8.044 L5B6 Given = c/...

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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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Functions of a random variable Given: p x ( ζ ) and f ( x ) Find: p f ( η ) f(x) x 8.044 L5B1
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A. Sketch f ( x ). Find where f ( x ) B. Integrate to find P f ( η ). C. Differentiate to find p f ( η ). 8.044 L5B2
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Example Intensity of light I = a E 2 p ( E )= 2 1 πσ 2 exp[ −E 2 / 2 σ 2 ] A I ε η a η a 0 η a ε 2 B ± η/a P I ( η )= η/a p E ( ζ ) 8.044 L5B3
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± g x a(y) b(y) da dy dy db dy dy g y dy d b ( y ) a ( y ) g ( y, x ) dx = dy g ( y, x = b ( y )) db ( y ) dy g ( y, x = a ( y )) da ( y ) dy + ± b ( y ) a ( y ) ∂g ( y, x ) ∂y dx 8.044 L5B4
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C In general 1 1 p I ( η )= 2 ηa p E ( ± η/a ) ( 1 1 2 ηa ) p E ( ± η/a ) = 1 1 2 ηa [ p E ( ± η/a )+ p E ( ± η/a )] In our particular case 1 I p ( I )= exp[ 2 πaσ 2 I 2 2 ] 1 2 3 4 5 I /a σ 2 a σ 2 p( I ) 2 π 0 1 2 3 4 8.044 L5B5
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Example Black Body Radiation p ( ν )= 1 2 ζ (3) ± ²³ ´ 1 / 2 . 404 1 ν 0 ( ν/ν 0 ) 2 exp[ ν/ν 0 ] 1 ν 0 = kT/h
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Unformatted text preview: T p( ) ( / ) ( / ) 2 e ( / ) 8.044 L5B6 Given = c/ and p ( ) A Find p ( ) B P ( ) = p ( ) d c / = c / c/ 8.044 L5B7 C In general p ( ) = ( c/ 2 ) p ( c/ ) In our case c 1 1 ( c/ ) 2 p ( ) = 2 2 . 404 0 exp[( c/ )] 1 Let 0 c/ , then 1 1 0 4 1 p ( ) = 2 . 404 0 exp[( / ) 1 ] 1 8.044 L5B8 1 As ( / ) 0 exp[( / ) 1 ] 1 e ( / ) 1 1 1 As ( / ) exp[( / ) 1 ] 1 (1+( / ) 1 1) ( / ) p( ) ( / )-1 ( / )-4 e ( / )-3 8.044 L5B9...
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lec5 - T p( ) ( / ) ( / ) 2 e ( / ) 8.044 L5B6 Given = c/...

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