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[(equation.5.11) (equation.5.14b)] +/Names [(equation.3.14a) 747 0 R (equation.3.14b) 748 0 R (equation.3.15) 749 0 R (equation.3.16) 755 0 R (equation.3.17) 756 0 R (equation.3.18) 757 0 R] +/Limits [(equation.3.14a) (equation.3.18)] >> endobj 1014 0 obj << -/Names [(equation.5.15) 901 0 R (equation.5.16) 906 0 R (equation.5.17) 907 0 R (equation.5.18) 908 0 R (equation.5.18a) 909 0 R (equation.5.18b) 910 0 R] -/Limits [(equation.5.15) (equation.5.18b)] +/Names [(equation.3.19) 758 0 R (equation.3.2) 711 0 R (equation.3.20) 760 0 R (equation.3.21) 761 0 R (equation.3.22) 768 0 R (equation.3.23) 769 0 R] +/Limits [(equation.3.19) (equation.3.23)] >> endobj 1015 0 obj << -/Names [(equation.5.19) 911 0 R (equation.5.2) 853 0 R (equation.5.20) 912 0 R (equation.5.3) 854 0 R (equation.5.4) 860 0 R (equation.5.5) 861 0 R] -/Limits [(equation.5.19) (equation.5.5)] +/Names [(equation.3.3) 712 0 R (equation.3.3a) 713 0 R (equation.3.3b) 714 0 R (equation.3.4) 715 0 R (equation.3.4a) 716 0 R (equation.3.4b) 717 0 R] +/Limits [(equation.3.3) (equation.3.4b)] >> endobj 1016 0 obj << -/Names [(equation.5.5a) 862 0 R (equation.5.5b) 863 0 R (equation.5.6) 864 0 R (equation.5.6a) 865 0 R (equation.5.6b) 866 0 R (equation.5.6c) 867 0 R] -/Limits [(equation.5.5a) (equation.5.6c)] +/Names [(equation.3.4c) 718 0 R (equation.3.5) 719 0 R (equation.3.6) 726 0 R (equation.3.7) 727 0 R (equation.3.7a) 728 0 R (equation.3.7b) 729 0 R] +/Limits [(equation.3.4c) (equation.3.7b)] >> endobj 1017 0 obj << -/Names [(equation.5.7) 868 0 R (equation.5.7a) 869 0 R (equation.5.7b) 870 0 R (equation.5.7c) 871 0 R (equation.5.8) 876 0 R (equation.5.8a) 877 0 R] -/Limits [(equation.5.7) (equation.5.8a)] +/Names [(equation.3.7c) 730 0 R (equation.3.8) 731 0 R (equation.3.9) 732 0 R (equation.4.1) 775 0 R (equation.4.10) 819 0 R (equation.4.11) 820 0 R] +/Limits [(equation.3.7c) (equation.4.11)] >> endobj 1018 0 obj << -/Names [(equation.5.8b) 878 0 R (equation.5.8c) 879 0 R (equation.5.9) 880 0 R (equation.6.1) 918 0 R (equation.6.1a) 919 0 R (equation.6.1b) 920 0 R] -/Limits [(equation.5.8b) (equation.6.1b)] +/Names [(equation.4.12) 827 0 R (equation.4.13) 828 0 R (equation.4.14) 829 0 R (equation.4.15) 830 0 R (equation.4.16) 840 0 R (equation.4.17) 841 0 R] +/Limits [(equation.4.12) (equation.4.17)] >> endobj 1019 0 obj << -/Names [(equation.6.1c) 921 0 R (equation.6.2) 922 0 R (equation.6.3) 923 0 R (equation.6.4) 929 0 R (equation.6.4a) 930 0 R (equation.6.4b) 931 0 R] -/Limits [(equation.6.1c) (equation.6.4b)] +/Names [(equation.4.18) 843 0 R (equation.4.18a) 844 0 R (equation.4.18b) 845 0 R (equation.4.18c) 846 0 R (equation.4.18d) 847 0 R (equation.4.2) 776 0 R] +/Limits [(equation.4.18) (equation.4.2)] >> endobj 1020 0 obj << -/Names [(equation.6.4c) 932 0 R (equation.6.4d) 933 0 R (equation.6.4e) 934 0 R (equation.6.4f) 935 0 R (equation.6.4g) 936 0 R (equation.6.5) 937 0 R] -/Limits [(equation.6.4c) (equation.6.5)] +/Names [(equation.4.3) 778 0 R (equation.4.3a) 780 0 R (equation.4.3b) 781 0 R (equation.4.3c) 782 0 R (equation.4.4) 798 0 R (equation.4.4a) 799 0 R] +/Limits [(equation.4.3) (equation.4.4a)] >> endobj 1021 0 obj << -/Names [(equation.6.6) 938 0 R (equation.6.7) 943 0 R (page.0) 444 0 R (page.1) 456 0 R (page.10) 654 0 R (page.11) 671 0 R] -/Limits [(equation.6.6) (page.11)] +/Names [(equation.4.4b) 800 0 R (equation.4.5) 801 0 R (equation.4.6) 803 0 R (equation.4.6a) 804 0 R (equation.4.6b) 805 0 R (equation.4.6c) 807 0 R] +/Limits [(equation.4.4b) (equation.4.6c)] >> endobj 1022 0 obj << -/Names [(page.12) 677 0 R (page.13) 687 0 R (page.14) 700 0 R (page.15) 715 0 R (page.16) 731 0 R (page.17) 745 0 R] -/Limits [(page.12) (page.17)] +/Names [(equation.4.6d) 808 0 R (equation.4.6e) 809 0 R (equation.4.7) 810 0 R (equation.4.8) 815 0 R (equation.4.8a) 816 0 R (equation.4.8b) 817 0 R] +/Limits [(equation.4.6d) (equation.4.8b)] >> endobj 1023 0 obj << -/Names [(page.18) 758 0 R (page.19) 765 0 R (page.2) 492 0 R (page.20) 788 0 R (page.21) 805 0 R (page.22) 817 0 R] -/Limits [(page.18) (page.22)] +/Names [(equation.4.9) 818 0 R (equation.5.1) 861 0 R (equation.5.10) 890 0 R (equation.5.11) 897 0 R (equation.5.12) 898 0 R (equation.5.13) 899 0 R] +/Limits [(equation.4.9) (equation.5.13)] >> endobj 1024 0 obj << -/Names [(page.23) 826 0 R (page.24) 851 0 R (page.25) 859 0 R (page.26) 875 0 R (page.27) 887 0 R (page.28) 897 0 R] -/Limits [(page.23) (page.28)] +/Names [(equation.5.14) 907 0 R (equation.5.14a) 908 0 R (equation.5.14b) 909 0 R (equation.5.15) 910 0 R (equation.5.16) 915 0 R (equation.5.17) 916 0 R] +/Limits [(equation.5.14) (equation.5.17)] >> endobj 1025 0 obj << -/Names [(page.29) 905 0 R (page.3) 538 0 R (page.30) 917 0 R (page.31) 928 0 R (page.32) 942 0 R (page.4) 582 0 R] -/Limits [(page.29) (page.4)] +/Names [(equation.5.18) 917 0 R (equation.5.18a) 918 0 R (equation.5.18b) 919 0 R (equation.5.19) 920 0 R (equation.5.2) 862 0 R (equation.5.20) 921 0 R] +/Limits [(equation.5.18) (equation.5.20)] >> endobj 1026 0 obj << -/Names [(page.5) 590 0 R (page.6) 595 0 R (page.7) 613 0 R (page.8) 633 0 R (page.9) 643 0 R (section*.3) 494 0 R] -/Limits [(page.5) (section*.3)] +/Names [(equation.5.3) 863 0 R (equation.5.4) 869 0 R (equation.5.5) 870 0 R (equation.5.5a) 871 0 R (equation.5.5b) 872 0 R (equation.5.6) 873 0 R] +/Limits [(equation.5.3) (equation.5.6)] >> endobj 1027 0 obj << -/Names [(section.1.1) 6 0 R (section.1.2) 10 0 R (section.1.3) 22 0 R (section.1.4) 34 0 R (section.1.5) 50 0 R (section.1.6) 54 0 R] -/Limits [(section.1.1) (section.1.6)] +/Names [(equation.5.6a) 874 0 R (equation.5.6b) 875 0 R (equation.5.6c) 876 0 R (equation.5.7) 877 0 R (equation.5.7a) 878 0 R (equation.5.7b) 879 0 R] +/Limits [(equation.5.6a) (equation.5.7b)] >> endobj 1028 0 obj << -/Names [(section.1.7) 66 0 R (section.1.8) 78 0 R (section.2.1) 106 0 R (section.2.2) 118 0 R (section.2.3) 138 0 R (section.3.1) 150 0 R] -/Limits [(section.1.7) (section.3.1)] +/Names [(equation.5.7c) 880 0 R (equation.5.8) 885 0 R (equation.5.8a) 886 0 R (equation.5.8b) 887 0 R (equation.5.8c) 888 0 R (equation.5.9) 889 0 R] +/Limits [(equation.5.7c) (equation.5.9)] >> endobj 1029 0 obj << -/Names [(section.3.2) 154 0 R (section.3.3) 166 0 R (section.3.4) 186 0 R (section.3.5) 202 0 R (section.3.6) 214 0 R (section.4.1) 246 0 R] -/Limits [(section.3.2) (section.4.1)] +/Names [(equation.6.1) 927 0 R (equation.6.1a) 928 0 R (equation.6.1b) 929 0 R (equation.6.1c) 930 0 R (equation.6.2) 931 0 R (equation.6.3) 932 0 R] +/Limits [(equation.6.1) (equation.6.3)] >> endobj 1030 0 obj << -/Names [(section.4.2) 274 0 R (section.4.3) 282 0 R (section.4.4) 298 0 R (section.5.1) 314 0 R (section.5.2) 350 0 R (section.5.3) 370 0 R] -/Limits [(section.4.2) (section.5.3)] +/Names [(equation.6.4) 938 0 R (equation.6.4a) 939 0 R (equation.6.4b) 940 0 R (equation.6.4c) 941 0 R (equation.6.4d) 942 0 R (equation.6.4e) 943 0 R] +/Limits [(equation.6.4) (equation.6.4e)] >> endobj 1031 0 obj << -/Names [(section.6.1) 402 0 R (section.6.2) 414 0 R (subsection.1.2.1) 14 0 R (subsection.1.2.2) 18 0 R (subsection.1.3.1) 26 0 R (subsection.1.3.2) 30 0 R] -/Limits [(section.6.1) (subsection.1.3.2)] +/Names [(equation.6.4f) 944 0 R (equation.6.4g) 945 0 R (equation.6.5) 946 0 R (equation.6.6) 947 0 R (equation.6.7) 952 0 R (page.0) 444 0 R] +/Limits [(equation.6.4f) (page.0)] >> endobj 1032 0 obj << -/Names [(subsection.1.4.1) 38 0 R (subsection.1.4.2) 42 0 R (subsection.1.4.3) 46 0 R (subsection.1.6.1) 58 0 R (subsection.1.6.2) 62 0 R (subsection.1.7.1) 70 0 R] -/Limits [(subsection.1.4.1) (subsection.1.7.1)] +/Names [(page.1) 456 0 R (page.10) 660 0 R (page.11) 678 0 R (page.12) 686 0 R (page.13) 696 0 R (page.14) 709 0 R] +/Limits [(page.1) (page.14)] >> endobj 1033 0 obj << -/Names [(subsection.1.7.2) 74 0 R (subsection.1.8.1) 82 0 R (subsection.1.8.2) 86 0 R (subsection.1.8.3) 90 0 R (subsection.1.8.4) 94 0 R (subsection.1.8.5) 98 0 R] -/Limits [(subsection.1.7.2) (subsection.1.8.5)] +/Names [(page.15) 724 0 R (page.16) 740 0 R (page.17) 754 0 R (page.18) 767 0 R (page.19) 774 0 R (page.2) 492 0 R] +/Limits [(page.15) (page.2)] >> endobj 1034 0 obj << -/Names [(subsection.2.1.1) 110 0 R (subsection.2.1.2) 114 0 R (subsection.2.2.1) 122 0 R (subsection.2.2.2) 126 0 R (subsection.2.2.3) 130 0 R (subsection.2.2.4) 134 0 R] -/Limits [(subsection.2.1.1) (subsection.2.2.4)] +/Names [(page.20) 797 0 R (page.21) 814 0 R (page.22) 826 0 R (page.23) 835 0 R (page.24) 860 0 R (page.25) 868 0 R] +/Limits [(page.20) (page.25)] >> endobj 1035 0 obj << -/Names [(subsection.2.3.1) 142 0 R (subsection.3.2.1) 158 0 R (subsection.3.2.2) 162 0 R (subsection.3.3.1) 170 0 R (subsection.3.3.2) 174 0 R (subsection.3.3.3) 178 0 R] -/Limits [(subsection.2.3.1) (subsection.3.3.3)] +/Names [(page.26) 884 0 R (page.27) 896 0 R (page.28) 906 0 R (page.29) 914 0 R (page.3) 538 0 R (page.30) 926 0 R] +/Limits [(page.26) (page.30)] >> endobj 1036 0 obj << -/Names [(subsection.3.3.4) 182 0 R (subsection.3.4.1) 190 0 R (subsection.3.4.2) 194 0 R (subsection.3.4.3) 198 0 R (subsection.3.5.1) 206 0 R (subsection.3.5.2) 210 0 R] -/Limits [(subsection.3.3.4) (subsection.3.5.2)] +/Names [(page.31) 937 0 R (page.32) 951 0 R (page.4) 582 0 R (page.5) 590 0 R (page.6) 595 0 R (page.7) 613 0 R] +/Limits [(page.31) (page.7)] >> endobj 1037 0 obj << -/Names [(subsection.3.6.1) 218 0 R (subsection.3.6.2) 222 0 R (subsection.3.6.3) 226 0 R (subsection.3.6.4) 230 0 R (subsection.3.6.5) 234 0 R (subsection.3.6.6) 238 0 R] -/Limits [(subsection.3.6.1) (subsection.3.6.6)] +/Names [(page.8) 633 0 R (page.9) 648 0 R (section*.3) 494 0 R (section.1.1) 6 0 R (section.1.2) 10 0 R (section.1.3) 22 0 R] +/Limits [(page.8) (section.1.3)] >> endobj 1038 0 obj << -/Names [(subsection.4.1.1) 250 0 R (subsection.4.1.2) 254 0 R (subsection.4.1.3) 258 0 R (subsection.4.1.4) 262 0 R (subsection.4.1.5) 266 0 R (subsection.4.1.6) 270 0 R] -/Limits [(subsection.4.1.1) (subsection.4.1.6)] +/Names [(section.1.4) 34 0 R (section.1.5) 50 0 R (section.1.6) 54 0 R (section.1.7) 66 0 R (section.1.8) 78 0 R (section.2.1) 106 0 R] +/Limits [(section.1.4) (section.2.1)] >> endobj 1039 0 obj << -/Names [(subsection.4.2.1) 278 0 R (subsection.4.3.1) 286 0 R (subsection.4.3.2) 290 0 R (subsection.4.3.3) 294 0 R (subsection.4.4.1) 302 0 R (subsection.4.4.2) 306 0 R] -/Limits [(subsection.4.2.1) (subsection.4.4.2)] +/Names [(section.2.2) 118 0 R (section.2.3) 138 0 R (section.3.1) 150 0 R (section.3.2) 154 0 R (section.3.3) 166 0 R (section.3.4) 186 0 R] +/Limits [(section.2.2) (section.3.4)] >> endobj 1040 0 obj << -/Names [(subsection.5.1.1) 318 0 R (subsection.5.1.2) 322 0 R (subsection.5.1.3) 326 0 R (subsection.5.1.4) 330 0 R (subsection.5.1.5) 334 0 R (subsection.5.1.6) 338 0 R] -/Limits [(subsection.5.1.1) (subsection.5.1.6)] +/Names [(section.3.5) 202 0 R (section.3.6) 214 0 R (section.4.1) 246 0 R (section.4.2) 274 0 R (section.4.3) 282 0 R (section.4.4) 298 0 R] +/Limits [(section.3.5) (section.4.4)] >> endobj 1041 0 obj << -/Names [(subsection.5.1.7) 342 0 R (subsection.5.1.8) 346 0 R (subsection.5.2.1) 354 0 R (subsection.5.2.2) 358 0 R (subsection.5.2.3) 362 0 R (subsection.5.2.4) 366 0 R] -/Limits [(subsection.5.1.7) (subsection.5.2.4)] +/Names [(section.5.1) 314 0 R (section.5.2) 350 0 R (section.5.3) 370 0 R (section.6.1) 402 0 R (section.6.2) 414 0 R (subsection.1.2.1) 14 0 R] +/Limits [(section.5.1) (subsection.1.2.1)] >> endobj 1042 0 obj << -/Names [(subsection.5.3.1) 374 0 R (subsection.5.3.2) 378 0 R (subsection.5.3.3) 382 0 R (subsection.5.3.4) 386 0 R (subsection.5.3.5) 390 0 R (subsection.5.3.6) 394 0 R] -/Limits [(subsection.5.3.1) (subsection.5.3.6)] +/Names [(subsection.1.2.2) 18 0 R (subsection.1.3.1) 26 0 R (subsection.1.3.2) 30 0 R (subsection.1.4.1) 38 0 R (subsection.1.4.2) 42 0 R (subsection.1.4.3) 46 0 R] +/Limits [(subsection.1.2.2) (subsection.1.4.3)] >> endobj 1043 0 obj << -/Names [(subsection.6.1.1) 406 0 R (subsection.6.1.2) 410 0 R (subsection.6.2.1) 418 0 R (subsection.6.2.2) 422 0 R (subsection.6.2.3) 426 0 R (subsection.6.2.4) 430 0 R] -/Limits [(subsection.6.1.1) (subsection.6.2.4)] +/Names [(subsection.1.6.1) 58 0 R (subsection.1.6.2) 62 0 R (subsection.1.7.1) 70 0 R (subsection.1.7.2) 74 0 R (subsection.1.8.1) 82 0 R (subsection.1.8.2) 86 0 R] +/Limits [(subsection.1.6.1) (subsection.1.8.2)] >> endobj 1044 0 obj << -/Names [(subsection.6.2.5) 434 0 R (subsubsection*.10) 833 0 R (subsubsection*.4) 644 0 R (subsubsection*.5) 646 0 R (subsubsection*.6) 750 0 R (subsubsection*.7) 768 0 R] -/Limits [(subsection.6.2.5) (subsubsection*.7)] +/Names [(subsection.1.8.3) 90 0 R (subsection.1.8.4) 94 0 R (subsection.1.8.5) 98 0 R (subsection.2.1.1) 110 0 R (subsection.2.1.2) 114 0 R (subsection.2.2.1) 122 0 R] +/Limits [(subsection.1.8.3) (subsection.2.2.1)] >> endobj 1045 0 obj << -/Names [(subsubsection*.8) 793 0 R (subsubsection*.9) 827 0 R] -/Limits [(subsubsection*.8) (subsubsection*.9)] +/Names [(subsection.2.2.2) 126 0 R (subsection.2.2.3) 130 0 R (subsection.2.2.4) 134 0 R (subsection.2.3.1) 142 0 R (subsection.3.2.1) 158 0 R (subsection.3.2.2) 162 0 R] +/Limits [(subsection.2.2.2) (subsection.3.2.2)] >> endobj 1046 0 obj << -/Kids [990 0 R 991 0 R 992 0 R 993 0 R 994 0 R 995 0 R] -/Limits [(Doc-Start) (equation.1.25)] +/Names [(subsection.3.3.1) 170 0 R (subsection.3.3.2) 174 0 R (subsection.3.3.3) 178 0 R (subsection.3.3.4) 182 0 R (subsection.3.4.1) 190 0 R (subsection.3.4.2) 194 0 R] +/Limits [(subsection.3.3.1) (subsection.3.4.2)] >> endobj 1047 0 obj << -/Kids [996 0 R 997 0 R 998 0 R 999 0 R 1000 0 R 1001 0 R] -/Limits [(equation.1.26) (equation.3.12a)] +/Names [(subsection.3.4.3) 198 0 R (subsection.3.5.1) 206 0 R (subsection.3.5.2) 210 0 R (subsection.3.6.1) 218 0 R (subsection.3.6.2) 222 0 R (subsection.3.6.3) 226 0 R] +/Limits [(subsection.3.4.3) (subsection.3.6.3)] >> endobj 1048 0 obj << -/Kids [1002 0 R 1003 0 R 1004 0 R 1005 0 R 1006 0 R 1007 0 R] -/Limits [(equation.3.12b) (equation.4.14)] +/Names [(subsection.3.6.4) 230 0 R (subsection.3.6.5) 234 0 R (subsection.3.6.6) 238 0 R (subsection.4.1.1) 250 0 R (subsection.4.1.2) 254 0 R (subsection.4.1.3) 258 0 R] +/Limits [(subsection.3.6.4) (subsection.4.1.3)] >> endobj 1049 0 obj << -/Kids [1008 0 R 1009 0 R 1010 0 R 1011 0 R 1012 0 R 1013 0 R] -/Limits [(equation.4.15) (equation.5.14b)] +/Names [(subsection.4.1.4) 262 0 R (subsection.4.1.5) 266 0 R (subsection.4.1.6) 270 0 R (subsection.4.2.1) 278 0 R (subsection.4.3.1) 286 0 R (subsection.4.3.2) 290 0 R] +/Limits [(subsection.4.1.4) (subsection.4.3.2)] >> endobj 1050 0 obj << -/Kids [1014 0 R 1015 0 R 1016 0 R 1017 0 R 1018 0 R 1019 0 R] -/Limits [(equation.5.15) (equation.6.4b)] +/Names [(subsection.4.3.3) 294 0 R (subsection.4.4.1) 302 0 R (subsection.4.4.2) 306 0 R (subsection.5.1.1) 318 0 R (subsection.5.1.2) 322 0 R (subsection.5.1.3) 326 0 R] +/Limits [(subsection.4.3.3) (subsection.5.1.3)] >> endobj 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>> startxref -308826 +310009 %%EOF diff --git a/document.tex b/document.tex index 10897ed..106047b 100644 --- a/document.tex +++ b/document.tex @@ -23,7 +23,7 @@ \title{Stochastische Signale und Systeme} \subtitle{Zusammenfassung Formeln} -\subsubtitle{Autor: Daniel Thiem - studium@daniel-thiem.de\\Version 0.9.7.2 - 26.09.2012} +\subsubtitle{Autor: Daniel Thiem - studium@daniel-thiem.de\\Version 0.9.8 - 26.09.2012} \maketitle \newpage @@ -107,6 +107,13 @@ P(X>a)&=1-F(a) \begin{equation} f(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{- \frac{1}{2} \left(\frac{t-\mu}{\sigma}\right)^2} \end{equation} +Erwartungswert und Varianz: +\begin{subequations} +\begin{align} +\E[f(x)]&=\mu \\ +\Var[f(x)]&=\sigma^2 +\end{align} +\end{subequations} \subsection{Rechteckverteilung} \begin{align} @@ -120,6 +127,13 @@ F(x) &= \begin{cases} 1 & x > b \end{cases} \end{align} +Erwartungswert und Varianz: +\begin{subequations} +\begin{align} +\E[f(x)]&=\frac{a+b}{2} \\ +\Var[f(x)]&=\frac{(b-a)^2}{12} +\end{align} +\end{subequations} \subsection{Exponentialverteilung} \begin{align} @@ -132,6 +146,13 @@ F(x) &= \begin{cases} 1-e^{-\lambda t} & x\geq 0 \end{cases} \end{align} +Erwartungswert und Varianz: +\begin{subequations} +\begin{align} +\E[f(x)]&=\frac{1}{\lambda} \\ +\Var[f(x)]&=\frac{1}{\lambda^2} +\end{align} +\end{subequations} \section{Formel von Bayes}