Consider a system consisting of N independent parts. An extensive variable X has an average vale...

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Consider a system consisting of N independent parts. An extensive variable X has an average vale (X) and moments ((AX)m), where AX = X – (X). - (a) Determine the exponent a for the odd moments: (1) (b) Construct the probability weight using all the odd moments and show that the they do not contribute to the distribution function as N→ ∞. A trick that may be helpful (and which is used in the standard derivation of the central-limit theorem) is to consider the variable Y = X/√N. note that ((AY)2) = O(1). ((AX)2m+1)1/(2m+1) (X) 1 Na Consider a system consisting of N independent parts. An extensive variable X has an average vale (X) and moments ((AX)m), where AX = X – (X). - (a) Determine the exponent a for the odd moments: (1) (b) Construct the probability weight using all the odd moments and show that the they do not contribute to the distribution function as N→ ∞. A trick that may be helpful (and which is used in the standard derivation of the central-limit theorem) is to consider the variable Y = X/√N. note that ((AY)2) = O(1). ((AX)2m+1)1/(2m+1) (X) 1 Na

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