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# PS1 - Chem 120B Problem Set 1 Due 1(i Show that the mean...

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Chem 120B Problem Set 1 Due: January 28, 2011 1. (i) Show that the mean square fluctuation in any quantity A can be written in terms of its mean and mean square values, ( ( δA ) 2 ) = ( A 2 ) − ( A ) 2 . (Here, as usual, the fluctuation in A is defined as δA = A − ( A ) .) (ii) Generalize this result to the case of two different fluctuation quantities, i.e., show that ( δA δB ) = ( AB ) − ( A )( B ) . 2. Consider two fluctuating variables x and y , whose statistics are described by a probability distribution p ( x, y ) . (i) If fluctuations in x and y are statistically independent, then their joint distribution p ( x, y ) factorizes, p ( x, y ) = f ( x ) g ( y ) , where f ( x ) and g ( y ) are the distributions of x and y , respectively. Explain why this factorization is implied by independence of x and y . (It is not necessary to give a very detailed account of your reasoning here, but it is important that you convince yourself of this fact.) (ii) The average of a function F ( x, y ) of these two variables can be written ( F ) = integraldisplay dx integraldisplay dy p ( x, y ) F ( x, y ) Consider a function F ( x, y ) = A ( x ) B ( y ) that depends separably on x and y . Show that the average of F also factorizes, ( F ) = ( A ( x ) )( B ( y ) ) . (iii) Using your results from parts (i) and (ii), show that statistical independence of x and y implies ( δx δy ) = 0 , where δx = x − ( x ) and δy = y − ( y ) are fluctuations of x and y about their mean values.

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PS1 - Chem 120B Problem Set 1 Due 1(i Show that the mean...

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