HW4-S10Soln - Homework 4 Solution Problem No 1(a)To compute...

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Homework 4 Solution Problem No. 1 (a)To compute the standard deviation x of N measurements x 1 , …,x N , one needs the sum 2 ( ) i x x . Prove that this sum can be rewritten as: 2 2 2 ( ) i i x x x N x . Solution: 2 2 2 ( ) ( 2 ) i i i x x x x x x but x is defined as: 1 i x x N putting this into the expression above 2 2 2 1 1 1 ( 2 ) ( 2 ) i i i i i i i x x x x x x x x x N N N now doing the outer summation: 2 1 1 1 2 i i i i i x x x N x x N N N simplifying 2 1 i i i x x x N but this is just: 2 2 i x N x (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 , i i x y , with the uncertainties in x negligible and those in y all equal.
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