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Astro 193 : HW 3
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The objective of this homework is to get you used to doing error propagation, both in the statistical-only case as well as where systematics are important. Beware that error propagation assumes symmetric error bars and Gaussian assumptions. Typically, people also assume that cross-terms in the covariance matrix are negligible. Check your assumptions before using this method.

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Astro 193 : HW 3
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g = a/b g/a = 1/b, g/b = -a/b² ²g = (1/b²)²a + (a²/b)²b ²g = (a²/b²)(²a/a²) + (a²/b²)(²b/b²) ²g = g² ( ²a/a² + ²b/b²) g/g = {²a/a² + ²b/b²}


Astro 193 : HW 3
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g = ln(a) g/a = 1/a ²g = (1/a²) ²a g = a/a


Astro 193 : HW 3
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g = ln(a)-ln(b) g/a = 1/a, g/b = -1/b ²g = (1/a²)²a + (1/b2)²b g = {²a/a² + ²b/b²}


Astro 193 : HW 3
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same result is obtained for ln(a/b), as it should, since a mathematical result is invariant to the process of derivation. g = ln(a)-ln(b) ln(a/b) g/a = 1/(a/b)(1/b) = (b/a)(1/b) = 1/a g/b = 1/(a/b) (-a/b2 ) = (b/a)(-a/b2) = -1/b ²g = (1/a²)²a + (1/b2)²b g = {²a/a² + ²b/b²}

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Astro 193 : HW 3
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There is no a priori reason to take the measurement from one instrument to be "better" than the other. Even the size of the uncertainty is not a definitive indicator, because a small uncertainty can be derived as a consequence of energy coverage. The uncertainty on the estimate is distributed as a Student's t distribution, because we are looking at combinations of Gaussians. For small degrees of freedom, this means the computed sigma should be inflated to match the appropriate enclosed probability mass. Degrees of freedom is a measure of how many sources of uncertainty there are in the calculation of a quantity. That is what accounts for the (N-1) in the denominator of the variance, because there is one less dof due to the previously calculated sample mean. Sometimes these dofs are used in the definition of a distribution (like the t, the chi^2, etc.) and are simply used as parameters without attached physical meaning. That is partly the case here -- the dof of the t distribution is the parameter that accounts for the inflation in the assumed Gaussian sigma.

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Astro 193 : HW 3
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NH = 3.07±0.02, 3.07±0.04, 3.18±0.04, 3.07±0.04, 2.93±0.06, 3.10±0.055, 2.90±0.035, 2.94±0.035, 2.74±0.025 = 1.84±0.01, 2.09±0.07, 2.06±0.015, 1.91±0.015, 1.92±0.015, 1.90±0.02, 2.28±0.15, 1.77±0.025, 1.91±0.025, 1.78±0.015, 1.84±0.015, 1.76±0.01 = 3.00 ;
NH

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= 0.15

< > = 1.92 ; = 0.17 WNH = 0.00165 ; BNH = 0.0179 ; TNH = 0.0215 ; dof
NH

= 9.39

W = 0.00253 ; B = 0.0236 ; T = 0.0281; dof = 13.29




T <>
individual measurements with error bars

B

W



inflation factor