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The chi-square statistic is m 2 χobs = (di − (GmL2 )i )2 /σi2 . 20) i=1 2 depends on the random measurement errors in d, it is itself a random variable. It Since χobs 2 has a χ 2 distribution with ν = m − n degrees can be shown that under our assumptions, χobs of freedom [22, 33]. The probability density function for the χ 2 distribution is fχ 2 (x) = 1 1 x 2 ν−1 e−x/2 . 5. The χ 2 test provides a statistical assessment of the assumptions that we 2 and compare it to the used in finding the least squares solution.

Plot the data and your fitted model, and list the parameters, ai obtained in your regression. Clearly, the correct solution has a1 = 1, and all other ai = 0. Explain why your answer differs. 7 NOTES AND FURTHER READING Linear regression is a major subfield within statistics, and there are literally hundreds of associated textbooks. Many of these references focus on applications of linear regression in the social sciences. In such applications, the primary focus is often on determining which variables have an effect on response variables of interest (rather than on estimating parameter values for a predetermined model).

Again, this problem can be solved by the method of Lagrange multipliers. 5 The Method of Backus and Gilbert 51 of the averaging kernel increases. There is a trade-off between stability of the solution and the width of the averaging kernel. The method of Backus and Gilbert produces an estimate of the model at a particular point x. ˆ It is possible to use the method to compute estimates on a grid of points x1 , x2 , . . , xn . However, since the averaging kernels at these points may not be well localized around the grid points and may overlap in complicated ways, this is not equivalent to the simple collocation method discussed earlier.