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Unformatted text preview: Physics 132 Homework 3 Consider a set of N equivalent measurements of a quantity x (each measurement being labeled with a subscript xi) which we assume contain random errors. Assume the usual normalized model for the probability distribution for each measurement: (x - ) 1 exp(- i 2 ) f ( xi ) = 2 2
2 Also, use the usual definition for the expectation value for a function P ( x1 , x2 ,..., xN ) which can depend on all of the measurements: P ... P( x1 , x2 ,..., xN ) ( f ( x1 ) f ( x2 )... f ( xN ) ) dx1dx2 ...dxN Show that: (x - x )
i =1 i N 2 = ( N - 1) 2 The average value of the measurements, x , is also defined in the usual way: x 1 N x
i =1 N i ________________________________________________________________________ Some help: Don't be intimidated by all the integrals notice that the expressions factorize into a product of N integrals over a single variable (do this with care, however!) and use the well known results: (x - ) f ( x)dx = 1 xf ( x)dx = 2 f ( x)dx = 2 From the these integrals you can quickly see also that: 2 2 2 x f ( x)dx = + ...
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This note was uploaded on 03/30/2008 for the course 029 132 taught by Professor Skiff during the Spring '08 term at University of Iowa.
- Spring '08