In other words equations 37 and 89 apply however if

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. In other words, equations 3.7, and 8.9 apply. However, if you are interested in knowing about both , you must include their covariance. In our example, the large negative covariance follows logically just from looking at a graph: if you fit some points, all of which lie at positive t , then a more negative derived slope will raise the y -intercept. Specifically, the large negative covariance means that positive departures of A 0 are associated with negative departures of A 1 . So even though the individual values δA 0 = +34 and δA 1 = +9 are acceptable, you cannot conclude that the pair of values ( δA 0 , δA 1 ) = (+34 , +9) is acceptable, because this pair has both positive. In contrast, what is acceptable here would be something like ( δA 0 , δA 1 ) = (+34 , 9). We stress that the acceptable ranges of values depend on what you are interested in. This is sort of like the observer’s influence in quantum mechanics. If you are interested in A 1 alone, then you can say A 1 = 4 ± 9 and, in making this statement, you have to realize that, as A 1 varies over this range, A 0 can vary over (formally, at least) the range ( ∞→−∞ ): you just don’t give a damn what happens to A 0 because you’re not interested. But the moment you become interested and restrict its possible range, that influences the possible range for A 1 , too. There is no simple relationship between the covariance matrix elements and the acceptable ranges. For two variables, the best way to express this is to construct the ellipses that define the loci of constant Δ χ 2 and present them on a graph with axes ( δa 0 , δa 1 ) as in BR Figure 11.2 or NR Figure 14.5.4. For three variables, these ellipses become ellipsoids; for four, they become four-dimensional volumes, etc. We illustrate these concepts for the ( a 1 , a 2 ) parameters in our numerical example. We sub- tracted 7.75 from all times so that the covariance would be small enough to illustrate the difference between the tangents to the ellipses and the end points of the ellipses. Contours are calculated as described in § 9.5 and are at Δ χ 2 = 1 and 2.3. The dashed horizontal and vertical lines are at δ a = ± σ a . First consider the pair of vertical lines, which are drawn at δa 1 = ± σ a 1 , where σ is the square root of the variance of the parameters as described in equations 3.7, 4.11, 8.9, and 8.10. If the datapoints were projected downward, i.e. look at the marginal pdf of δa 1 by taking small strips of δa 1 and integrating over δa 2 , the marginal pdf of δa 1 is Gaussian; ditto for the other coordinate. Thus, 68% of the points lie between these dashed lines. This is what we mean by the phrase “being interested in knowing about a 1 without regard to a 2 ”. If we allow a 2 to vary so as to minimize χ 2
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– 33 – Fig. 9.1.— Illustrating the meaning of variance and covariance between ( a 1 , a 2) for our numerical example. See text for discussion.
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