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Two events A1 and A2 are said to be mutually exclusive, or disjoint, when A1 and A2 have no common outcomes, that is A1 A2 = . For any two such disjoint events A1 and A2 , P(A1 A2 ) = P(A1 ) + P(A2 ) Now we are ready to state Axiom 3, which is the general case for k disjoint events. Axiom 3: Given k mutually exclusive events A1 , A2 , A3 , . . . , Ak , the probability that at least one of the events occurring (i.e., the probability of the union of the k events) is equal to the sum of the individual probabilities: P(A1 A2 A3 . . . Ak ) = k P(Ai ) i=1 On the other hand, when two events A1 and A2 are not mutually exclusive, (i.e., A1 A2 = ), then we must use the following formula when calculating the union of the two events: P(A1 A2 ) = P(A1 ) + P(A2 ) P(A1 A2 ) (8.4)

If a monic polynomial g(x) divides (x" - l), then it can be used to generate an ideal: I = (g(x)). In the ring IF, [x]/(x" - l), different ideals can be obtainedby selecting different divisors g(x) of X" - 1.

6. curl v [sin z, tan2 x 1, cos y] 8. A proof follows by calculation directly from the definitions. 10. curl v [0, 0, 2y], div v 0, incompressible; paths are obtained as follows. We have v r [x , y, z] [ y 2, 4, 0]. y2

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whichresults in ~ ( X O 1Ir) = 0.00295. Similarly,j ( x 0 = OJr)= 4.41 x lo-". After normalizing = P ( x 0 = 1Ir) = 0.999. Table 14.2 shows the posterior these, we find P ( x 0 = Olr) = 1.49 x bit probabilities (to three decimal places, so there is some roundoff in the probabilities near 1) and the posterior estimate of the bit sequence. Note that the estimated bit sequence matches the input sequence.

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June 12, 2004

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Figure 12.5 LTS for JITTER ( 2 ) .

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Figure 10.6 (a) Seismic results and (b) interpretation of depth-migrated ANCORP re ection data, including onshore wide-angle and receiver function results of Yuan et al. (2000) merged with offshore results of CINCA experiment (Patzwahl et al., 1999) (image provided by O. Oncken and modi ed from ANCORP Working Group, 2003, by permission of the American Geophysical Union. Copyright 2003 American Geophysical Union). Location of section shown in Fig. 10.1a. Inset in (b) shows an alternative interpretation of the slab geometry and seismicity. ALVZ, Altiplano low velocity zone; QBBS, Quebrada Blanca Bright Spot.

This chapter discusses the following: r r r r r The principle of Operational Quali cation (OQ) What needs to be included in OQ What happens if the supplier has done OQ The OQ plan and report OQ testing

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