hw4soln - Problem No. 1 (a)To compute the standard...

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Problem No. 1 (a)To compute the standard deviation x V of N measurements x 1 , ±,x N , one needs the sum 2 () i xx ± ¦ . Prove that this sum can be rewritten as: ²³ 2 22 ii x N x ªº ± ± ¬¼ ¦¦ . Solution: 2 ( 2 ) i x x x x ± ´ but x is defined as: 1 i N ¦ putting this into the expression above 2 11 1 (2 ) ( 2 ) i i i i i x x x x x x x NN N ±´ ± ´ ¦ ¦ ¦ now doing the outer summation: 2 1 2 i i i x N x x N ¦ ¦ ¦ simplifying 2 1 i x N ± ¦ but this is just: 2 2 i x Nx ± ¦ (b) Suppose two variables x and y are known to satisfy the relation y = Bx; i.e., they lie on a straight line that is known to pass through the origin. Suppose further that you have N measurements , xy , with the uncertainties in x negligible and those in y all equal. Show that the least-squares best estimate for B is 2 i x y B x ¦ ¦ .
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Solution: The numerator of the least squares sum is: 2 () ii yB x ± ¦ Now taking the derivative with respect to B 2 2 (( ) ) i i i x y Bx x B w ± ± ± w ¦¦ rewriting gives: 2 2( ) i yx B x ±± setting this equal to zero and solving for B 2 2 i i y x B x B x ² ¦ ¦
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Problem No. 2
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This note was uploaded on 12/26/2011 for the course ME 365 taught by Professor Merkle during the Fall '07 term at Purdue University-West Lafayette.

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hw4soln - Problem No. 1 (a)To compute the standard...

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