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Appendix A: Mathematical Formulas Appendix A: Mathematical Formulas Logarithms Inab = \na -\-\nb', \na/b = \na— \nb; \na^=b\na Geometric Series If Irl < 1, we find that oo oo E ar^ = ; y^akr^ = r; ^=0 A:=o y^ n Z-^ 1 -r ^^ 1 -r ^=1 k=0 Limits UHospitaVs rule. Suppose that \\vOix->XQ fix) = lim;c-^A:o sM = 0* or that liirix-^xo Z^-^) = l™x^jco SM — =t^^- Then, under some conditions, we can write that y fix) ^. fix) um = hm . x-^XQ g{x) x^xo g^{x) If the functions f\x) and g\x) satisfy the same conditions as f(x) and g(x), we can repeat the process. Moreover, the constant XQ may be equal to ±oo. Derivatives Derivative of a quotient: d fix) gix)fix) - fix)g\x) = 7^ 11 gix) ^0. dx gix) g^ix) 340 Appendix A: Mathematical Formulas Remark. This formula can also be obtained by differentiating the product f(x)h(x), where/Z(A:) := l/g(x). Chain rule. If w = ^(jc), then we have: d d du , , ^fiu) = —f(u) . — = f\u)g\x) = AgixWix). dx du dx For example, if w = jc^, we calculate — exp(M^) = exp(w^)2w • 2x = 4x^ expU"*). dx Integrals Integration by parts: j udv = uv — I vdu. Integration by substitution. If the inverse function g~^ exists, then we can write that rh rd j f(x)dx = j f[g(y)]g{y)dy, where a = g(c) ^ c = g ^(a) and b = g(d) <^ d = g ^(b). For example, / e''^'dx-^=^' f 2yeydy=2yey\l-2 f e^ dy = 2e^-\-2. Jo Jo Jo In this example, y = g~\x) = x^^^ and JC = g(y) = y^. Appendix B: Quantiles of the Sampling Distributions Gaussian distribution a 0.25 0.10 0.05 0.025 0.01 0.005 0.001 0.0005 Za 0.674 1.282 1.645 1.960 2.326 2.576 3.090 3.291 Student distribution n 1 2 3 4 5 6 7 8 ^0.025,n 12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306 t0.05,n 6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 n 9 10 15 20 25 30 40 oo tom5,n 2.262 2.228 2.131 2.086 2.060 2.042 2.021 1.960 t0.05,n 1.833 1.812 1.753 1.725 1.708 1.697 1.684 1.645 Chi-square distribution n 1 2 3 4 5 6 7 8 9 ^0.025,« 5.02 7.38 9.35 11.14 12.83 14.45 16.01 17.53 19.02 ^0.05,n 3.84 5.99 7.81 9.49 11.07 12.59 14.07 15.51 16.92 ^0.95,n 0+ 0.10 0.35 0.71 1.15 1.64 2.17 2.73 3.33 ^0.975,n 0+ 0.05 0.22 0.48 0.83 1.24 1.69 2.18 2.70 342 Appendix B: Quantiles of the Sampling Distributions Chi-square distribution (continued) n 10 15 20 25 30 40 50 100 -^0.025,M 20.48 27.49 34.17 40.65 46.98 59.34 71.42 129.56 ^0.05,^ 18.31 25.00 31.41 37.65 43.77 55.76 67.50 124.34 ^0.95,/I 3.94 7.26 10.85 14.61 18.49 26.51 34.76 77.93 3.25 6.27 9.59 13.12 16.79 24.43 32.36 74.22 ^Q.915,n Fisher distribution n 1 2 3 4 5 6 7 8 f^OX)25,n,n 647.8 39.00 15.44 9.60 7.15 5.82 4.99 4.43 f^{).i)5,n,n 161.4 19.00 9.28 6.39 5.05 4.28 3.79 3.44 n 9 10 15 20 30 60 120 oo ^0.025,/7,n 4.03 3.72 2.86 2.46 2.07 1.67 1.43 1.00 f^0X)5,n,n 3.18 2.98 2.40 2.12 1.84 1.53 1.35 1.00 Values of F().()25.^,,n2 n2\n\ 1 2 3 4 5 6 7 8 9 10 11 12 1 648 800 864 900 922 937 948 957 963 969 973 977 2 38.5 39.0 39.2 39.2 39.3 39.3 39.4 39.4 39.4 39.4 39.4 39.4 3 17.4 16.0 15.4 15.1 14.9 14.7 14.6 14.5 14.5 14.4 14.4 14.3 4 12.2 10.6 9.98 9.60 9.36 9.20 9.07 8.98 8.90 8.84 8.79 8.75 5 10.0 8.43 7.76 7.39 7.15 6.98 6.85 6.76 6.68 6.62 6.57 6.52 6 8.81 7.26 6.60 6.23 5.99 5.82 5.70 5.60 5.52 5.46 5.41 5.37 1 7 8.07 6.54 5.89 5.52 5.29 5.12 4.99 4.90 4.82 4.76 4.71 4.67 8 7.57 6.06 5.42 5.05 4.82 4.65 4.53 4.43 4.36 4.30 4.24 4.20 9 7.21 5.71 5.08 4.72 4.48 4.32 4.20 4.10 4.03 3.96 3.91 3.87 10 6.94 5.46 4.83 4.47 4.24 4.07 3.95 3.85 3.78 3.72 3.66 3.62 11 6.72 5.26 4.63 4.28 4.04 3.88 3.76 3.66 3.59 3.53 3.47 3.43 12 6.55 5.10 4.47 4.12 3.89 3.73 3.61 3.51 3.44 3.37 3.32 3.28 1 Appendix B: Quantiles of the Sampling Distributions 343 Values of Fo.o5,n 1,^2 n2\n\ 1 2 3 4 5 6 7 8 9 10 11 12 1 161 200 216 225 230 234 237 239 241 242 243 244 2 18.5 19.0 19.2 19.2 19.3 19.3 19.4 19.4 19.4 19.4 19.4 19.4 3 10.1 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8.79 8.76 8.74 4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96 5.94 5.91 5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.74 4.70 4.68 6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 4.06 4.03 4.00 7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3.64 3.60 3.57 8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35 3.31 3.28 9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3.14 3.10 3.07 10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.98 2.94 2.91 11 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90 2.85 2.82 2.79 12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75 2.72 2.69 Appendix C: Classification of the Exercises The list of exercises (among the multiple choice questions, problems or supplemen­ tary exercises) that can be done after having read each section is given below. The symbol Im (respectively Is), for example, denotes the multiple choice question (resp. supplementary exercise) no. 1 of the chapter in question. Chapter 2 2.1: Im, 6m, 12m, 19m, 32m 22: 2; 8m, 13m, 36m, 46m 23: 3, 8, 11, 18, 20; 2m, 7m, 1 Im, 14m, 15m, 21m, 24m, 29m, 34m 2.4: 1,5,24; 17m, 23m, 31m, 39m, 40m, 44m, 48m, 50m 2.5: all the others Chapter 3 3.1: - 3.2: 3m, 15m, 52m 33: 1,2,10,11,42; 4m, 16m, 28m, 34m, 35m, 36m, 53m 3.4: 12,19, 31, 34; Im, 12m, 17m, 18m, 29m, 37m, 56m, 79m 3.5: 14,27, 38; 6m, 19m, 20m, 41m, 58m, 64m, 80m 3.6: 6,35; 21m, 40m, 47m 3.7: 3, 4, 5, 7, 9, 15, 16, 17, 20, 22, 23, 25, 28, 29, 32, 37, 39, 43, 46, 47; 2m, 5m, 22m, 23m, 30m, 32m, 38m, 39m, 45m, 46m, 48m, 49m, 54m, 55m, 57m, 59m, 66m, 67m, 68m, 72m, 73m, 74m, 76m, 77m, 78m 3.8: 8,13, 18, 21, 24, 26, 30, 33, 36, 40, 41, 44, 45; 7m, 8m, 9m, 10m, 11m, 13m, 14m, 24m, 25m, 26m, 27m, 31m, 33m, 42m, 50m, 60m, 62m, 63m, 70m 3.9: 43m, 44m, 51m, 61m, 65m, 69m, 71m, 75m, 81m Chapter 4 4.1: - 4.2: 1, 2,12; 3m, 4m, 9m, 17m, 20m, 21m, 26m, 31m, 32m 43: 3,14; 10m, 22m, 33m, 40m 346 Appendix C: Classification of the Exercises 4.4:11m 4.5:4,5,19; 12m, 28m, 34m 4.6: 6, 16, 22, 24, 25, 27, 29; Im, 5m, 23m, 27m, 35m, 42m 4.7: 15, 17; 6m, 24m 4.8:7,20,21; 13m, 30m, 47m 4.9: 8, 9, 28; 7m, 18m, 29m, 36m 4.10: 16m 4.11: 10, 11, 12, 13, 18, 23,26; 2m, 8m, 14m, 15m, 19m, 25m, 37m, 38m, 39m, 41m, 43m, 44m, 45m, 46m, 48m Chapter 5 5.1: - 5.2: 3; 2m, 6m, 16m, 19m 53: 8; 3m, 11m, 14m, 23m, 27m, 29m 5.4: 1,4; lm,4m,7m, 17m, 20m, 24m, 25m, 26m, 28m, 30m 5.5: 7; 5m, 12m, 18m 5.6: 8m, 21m 5.7: 2, 5, 6; 9m, 10m, 13m, 15m, 22m Chapter 6 6.1: 1,6,8; 2m, 7m, 11m, 13m, 15m 6.2: 2,5,7; Im, 3m, 4m, 9m, 12m, 14m, 23m; Is, 2s, 4s 63: 3,4, 9; 5m, 6m, 8m, 10m, 18m, 21m, 25m, 27m 6.4: 10-19; 16m, 17m, 19m, 20m, 22m, 24m, 26m, 28m; 3s, 5s-l Is Chapter 7 7.1: 2m 7.2: Is 73:2s, 17s 7.4: 1; 14s 7.5: 3s, 5s 7.6: 2,3,4,5; Im, 4m; 4s, 6s, 7s, 9s, lis, 12s, 13s, 15s, 16s 7.7: 3m; 8s, 10s Appendix D: Answers to the Multiple Choice Questions Chapter 2 1 a; 2 d; 3 b; 4 d; 5 c; 6 c; 7 a; 8 c; 9 e; 10 c; 11 d; 12 c; 13 b; 14 a; 15 d; 16 c; 17 d; 18 c,e,b,b,c,d; 19 b,e; 20 d,c; 21 d,e; 22 d,a; 23 e,c; 24 a,b; 25 d,a; 26 b,c; 27 b,c,c,e; 28 d,a,a,c; 29 a,c,b; 30 e,b,a; 31 e,d,c; 32 b; 33 b; 34 d,c; 35 c,e,a,a; 36 c,a; 37 e,a,c,e; 38 b,d; 39 d,e; 40 c,b; 41 b,a,c,d; 42 c,e; 43 e,a; 44 d,d; 45 b,a; 46 e,c; 47 b,b; 48 d,d; 49 e,a; 50 b,d,a,e; 51 c,b,d,e.
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