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# lec7 - MIT OpenCourseWare http/ocw.mit.edu 8.044...

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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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Sums of Random Variables Consider n RVs x i and let s n x i . i =1 If the RVs are statistically independent, then <s> = Var( s )= i i <x i > Var( x i ) 8.044 L7B1
The individual p ( x i ) could be quite diﬀerent Both continuous and discrete RVs could be present True for any n Even if one RV dominates the sum 8.044 L7B2

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± ± Results have a special meaning when 1) The means are ﬁnite (=0) 2) The variances are ﬁnite (= ) 3) No subset dominates the sum 4) n is large width n 1 p(s) mean n n s 8.044 L7B3
Given p ( x, y ), ﬁnd p ( s x + y ) A dx y = α− x x+y = α x y α α B P s ( α )= ± ± α ζ p x,y ( ζ, η ) −∞ −∞ 8.044 L7B4

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p s ( α )= ± p x,y ( ζ, α ζ ) −∞ This is a general result; x and y need not be S.I. Application to the Jointly Gaussian RVs in Section 2
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lec7 - MIT OpenCourseWare http/ocw.mit.edu 8.044...

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