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(theorem.1.32) 583 0 R (theorem.1.33) 584 0 R (theorem.1.34) 585 0 R] +/Limits [(theorem.1.3) (theorem.1.34)] >> endobj 1050 0 obj << -/Names [(theorem.2.34) 649 0 R (theorem.2.35) 650 0 R (theorem.2.36) 651 0 R (theorem.2.37) 652 0 R (theorem.2.38) 654 0 R (theorem.2.39) 655 0 R] -/Limits [(theorem.2.34) (theorem.2.39)] +/Names [(theorem.1.35) 586 0 R (theorem.1.36) 587 0 R (theorem.1.37) 589 0 R (theorem.1.38) 590 0 R (theorem.1.39) 591 0 R (theorem.1.4) 530 0 R] +/Limits [(theorem.1.35) (theorem.1.4)] >> endobj 1051 0 obj << -/Names [(theorem.2.4) 608 0 R (theorem.2.40) 656 0 R (theorem.2.41) 657 0 R (theorem.2.42) 664 0 R (theorem.2.43) 665 0 R (theorem.2.44) 666 0 R] -/Limits [(theorem.2.4) (theorem.2.44)] +/Names [(theorem.1.40) 596 0 R (theorem.1.41) 597 0 R (theorem.1.42) 598 0 R (theorem.1.43) 599 0 R (theorem.1.44) 600 0 R (theorem.1.45) 601 0 R] +/Limits [(theorem.1.40) (theorem.1.45)] >> endobj 1052 0 obj << -/Names [(theorem.2.45) 667 0 R (theorem.2.46) 670 0 R (theorem.2.47) 671 0 R (theorem.2.48) 672 0 R (theorem.2.49) 673 0 R (theorem.2.5) 609 0 R] -/Limits [(theorem.2.45) (theorem.2.5)] +/Names [(theorem.1.46) 602 0 R (theorem.1.47) 603 0 R (theorem.1.5) 531 0 R (theorem.1.6) 532 0 R (theorem.1.7) 533 0 R (theorem.1.8) 534 0 R] +/Limits [(theorem.1.46) (theorem.1.8)] >> endobj 1053 0 obj << -/Names [(theorem.2.50) 674 0 R (theorem.2.51) 675 0 R (theorem.2.52) 676 0 R (theorem.2.53) 679 0 R (theorem.2.54) 687 0 R (theorem.2.55) 694 0 R] -/Limits [(theorem.2.50) (theorem.2.55)] +/Names [(theorem.1.9) 535 0 R (theorem.2.1) 609 0 R (theorem.2.10) 620 0 R (theorem.2.100) 768 0 R (theorem.2.101) 769 0 R (theorem.2.102) 778 0 R] +/Limits [(theorem.1.9) (theorem.2.102)] >> endobj 1054 0 obj << -/Names [(theorem.2.56) 695 0 R (theorem.2.57) 696 0 R (theorem.2.58) 697 0 R (theorem.2.59) 698 0 R (theorem.2.6) 610 0 R (theorem.2.60) 699 0 R] -/Limits [(theorem.2.56) (theorem.2.60)] +/Names [(theorem.2.103) 780 0 R (theorem.2.104) 781 0 R (theorem.2.105) 782 0 R (theorem.2.106) 783 0 R (theorem.2.107) 784 0 R (theorem.2.108) 785 0 R] +/Limits [(theorem.2.103) (theorem.2.108)] >> endobj 1055 0 obj << -/Names [(theorem.2.61) 700 0 R (theorem.2.62) 701 0 R (theorem.2.63) 702 0 R (theorem.2.64) 703 0 R (theorem.2.65) 704 0 R (theorem.2.66) 705 0 R] -/Limits [(theorem.2.61) (theorem.2.66)] +/Names [(theorem.2.11) 621 0 R (theorem.2.12) 622 0 R (theorem.2.13) 628 0 R (theorem.2.14) 629 0 R (theorem.2.15) 630 0 R (theorem.2.16) 631 0 R] +/Limits [(theorem.2.11) (theorem.2.16)] >> endobj 1056 0 obj << -/Names [(theorem.2.67) 713 0 R (theorem.2.68) 714 0 R (theorem.2.69) 715 0 R (theorem.2.7) 611 0 R (theorem.2.70) 716 0 R (theorem.2.71) 717 0 R] -/Limits [(theorem.2.67) (theorem.2.71)] +/Names [(theorem.2.17) 632 0 R (theorem.2.18) 633 0 R (theorem.2.19) 634 0 R (theorem.2.2) 610 0 R (theorem.2.20) 635 0 R (theorem.2.21) 636 0 R] +/Limits [(theorem.2.17) (theorem.2.21)] >> endobj 1057 0 obj << -/Names [(theorem.2.72) 718 0 R (theorem.2.73) 719 0 R (theorem.2.74) 720 0 R (theorem.2.75) 721 0 R (theorem.2.76) 722 0 R (theorem.2.77) 723 0 R] -/Limits [(theorem.2.72) (theorem.2.77)] +/Names [(theorem.2.22) 637 0 R (theorem.2.23) 638 0 R (theorem.2.24) 639 0 R (theorem.2.25) 640 0 R (theorem.2.26) 641 0 R (theorem.2.27) 648 0 R] +/Limits [(theorem.2.22) (theorem.2.27)] >> endobj 1058 0 obj << -/Names [(theorem.2.78) 724 0 R (theorem.2.79) 725 0 R (theorem.2.8) 612 0 R (theorem.2.80) 726 0 R (theorem.2.81) 727 0 R (theorem.2.82) 737 0 R] -/Limits [(theorem.2.78) (theorem.2.82)] +/Names [(theorem.2.28) 649 0 R (theorem.2.29) 650 0 R (theorem.2.3) 613 0 R (theorem.2.30) 651 0 R (theorem.2.31) 652 0 R (theorem.2.32) 653 0 R] +/Limits [(theorem.2.28) (theorem.2.32)] >> endobj 1059 0 obj << -/Names [(theorem.2.83) 738 0 R (theorem.2.84) 739 0 R (theorem.2.85) 740 0 R (theorem.2.86) 741 0 R (theorem.2.87) 742 0 R (theorem.2.88) 743 0 R] -/Limits [(theorem.2.83) (theorem.2.88)] +/Names [(theorem.2.33) 654 0 R (theorem.2.34) 655 0 R (theorem.2.35) 656 0 R (theorem.2.36) 657 0 R (theorem.2.37) 658 0 R (theorem.2.38) 660 0 R] +/Limits [(theorem.2.33) (theorem.2.38)] >> endobj 1060 0 obj << -/Names [(theorem.2.89) 744 0 R (theorem.2.9) 613 0 R (theorem.2.90) 745 0 R (theorem.2.91) 746 0 R (theorem.2.92) 749 0 R (theorem.2.93) 750 0 R] -/Limits [(theorem.2.89) (theorem.2.93)] +/Names [(theorem.2.39) 661 0 R (theorem.2.4) 614 0 R (theorem.2.40) 662 0 R (theorem.2.41) 663 0 R (theorem.2.42) 670 0 R (theorem.2.43) 671 0 R] +/Limits [(theorem.2.39) (theorem.2.43)] >> endobj 1061 0 obj << -/Names [(theorem.2.94) 751 0 R (theorem.2.95) 757 0 R (theorem.2.96) 758 0 R (theorem.2.97) 759 0 R (theorem.2.98) 760 0 R (theorem.2.99) 761 0 R] -/Limits [(theorem.2.94) (theorem.2.99)] +/Names [(theorem.2.44) 672 0 R (theorem.2.45) 673 0 R (theorem.2.46) 676 0 R (theorem.2.47) 677 0 R (theorem.2.48) 678 0 R (theorem.2.49) 679 0 R] +/Limits [(theorem.2.44) (theorem.2.49)] >> endobj 1062 0 obj << -/Names [(theorem.3.1) 791 0 R (theorem.3.10) 815 0 R (theorem.3.11) 822 0 R (theorem.3.12) 823 0 R (theorem.3.13) 826 0 R (theorem.3.14) 827 0 R] -/Limits [(theorem.3.1) (theorem.3.14)] +/Names [(theorem.2.5) 615 0 R (theorem.2.50) 680 0 R (theorem.2.51) 681 0 R (theorem.2.52) 682 0 R (theorem.2.53) 685 0 R (theorem.2.54) 693 0 R] +/Limits [(theorem.2.5) (theorem.2.54)] >> endobj 1063 0 obj << -/Names [(theorem.3.15) 828 0 R (theorem.3.16) 829 0 R (theorem.3.17) 830 0 R (theorem.3.18) 831 0 R (theorem.3.19) 832 0 R (theorem.3.2) 792 0 R] -/Limits [(theorem.3.15) (theorem.3.2)] +/Names [(theorem.2.55) 700 0 R (theorem.2.56) 701 0 R (theorem.2.57) 702 0 R (theorem.2.58) 703 0 R (theorem.2.59) 704 0 R (theorem.2.6) 616 0 R] +/Limits [(theorem.2.55) (theorem.2.6)] >> endobj 1064 0 obj << -/Names [(theorem.3.20) 833 0 R (theorem.3.21) 834 0 R (theorem.3.22) 847 0 R (theorem.3.23) 852 0 R (theorem.3.24) 853 0 R (theorem.3.25) 854 0 R] -/Limits [(theorem.3.20) (theorem.3.25)] +/Names [(theorem.2.60) 705 0 R (theorem.2.61) 706 0 R (theorem.2.62) 707 0 R (theorem.2.63) 708 0 R (theorem.2.64) 709 0 R (theorem.2.65) 710 0 R] +/Limits [(theorem.2.60) (theorem.2.65)] >> endobj 1065 0 obj << -/Names [(theorem.3.26) 855 0 R (theorem.3.27) 856 0 R (theorem.3.28) 857 0 R (theorem.3.29) 858 0 R (theorem.3.3) 796 0 R (theorem.3.30) 864 0 R] -/Limits [(theorem.3.26) (theorem.3.30)] +/Names [(theorem.2.66) 711 0 R (theorem.2.67) 719 0 R (theorem.2.68) 720 0 R (theorem.2.69) 721 0 R (theorem.2.7) 617 0 R (theorem.2.70) 722 0 R] +/Limits [(theorem.2.66) (theorem.2.70)] >> endobj 1066 0 obj << -/Names [(theorem.3.31) 865 0 R (theorem.3.32) 866 0 R (theorem.3.33) 867 0 R (theorem.3.34) 868 0 R (theorem.3.35) 869 0 R (theorem.3.36) 872 0 R] -/Limits [(theorem.3.31) (theorem.3.36)] +/Names [(theorem.2.71) 723 0 R (theorem.2.72) 724 0 R (theorem.2.73) 725 0 R (theorem.2.74) 726 0 R (theorem.2.75) 727 0 R (theorem.2.76) 728 0 R] +/Limits [(theorem.2.71) (theorem.2.76)] >> endobj 1067 0 obj << -/Names [(theorem.3.37) 875 0 R (theorem.3.38) 876 0 R (theorem.3.39) 877 0 R (theorem.3.4) 802 0 R (theorem.3.40) 878 0 R (theorem.3.41) 879 0 R] -/Limits [(theorem.3.37) (theorem.3.41)] +/Names [(theorem.2.77) 729 0 R (theorem.2.78) 730 0 R (theorem.2.79) 731 0 R (theorem.2.8) 618 0 R (theorem.2.80) 732 0 R (theorem.2.81) 733 0 R] +/Limits [(theorem.2.77) (theorem.2.81)] >> endobj 1068 0 obj << -/Names [(theorem.3.42) 880 0 R (theorem.3.43) 881 0 R (theorem.3.44) 882 0 R (theorem.3.45) 883 0 R (theorem.3.46) 889 0 R (theorem.3.47) 890 0 R] -/Limits [(theorem.3.42) (theorem.3.47)] +/Names [(theorem.2.82) 743 0 R (theorem.2.83) 744 0 R (theorem.2.84) 745 0 R (theorem.2.85) 746 0 R (theorem.2.86) 747 0 R (theorem.2.87) 748 0 R] +/Limits [(theorem.2.82) (theorem.2.87)] >> endobj 1069 0 obj << -/Names [(theorem.3.48) 891 0 R (theorem.3.49) 892 0 R (theorem.3.5) 803 0 R (theorem.3.50) 893 0 R (theorem.3.51) 894 0 R (theorem.3.52) 895 0 R] -/Limits [(theorem.3.48) (theorem.3.52)] +/Names [(theorem.2.88) 749 0 R (theorem.2.89) 750 0 R (theorem.2.9) 619 0 R (theorem.2.90) 751 0 R (theorem.2.91) 752 0 R (theorem.2.92) 755 0 R] +/Limits [(theorem.2.88) (theorem.2.92)] >> endobj 1070 0 obj << -/Names [(theorem.3.53) 896 0 R (theorem.3.54) 897 0 R (theorem.3.55) 898 0 R (theorem.3.56) 899 0 R (theorem.3.57) 900 0 R (theorem.3.58) 901 0 R] -/Limits [(theorem.3.53) (theorem.3.58)] +/Names [(theorem.2.93) 756 0 R (theorem.2.94) 757 0 R (theorem.2.95) 763 0 R (theorem.2.96) 764 0 R (theorem.2.97) 765 0 R (theorem.2.98) 766 0 R] +/Limits [(theorem.2.93) (theorem.2.98)] >> endobj 1071 0 obj << -/Names [(theorem.3.59) 902 0 R (theorem.3.6) 804 0 R (theorem.3.60) 903 0 R (theorem.3.61) 904 0 R (theorem.3.62) 910 0 R (theorem.3.63) 911 0 R] -/Limits [(theorem.3.59) (theorem.3.63)] +/Names [(theorem.2.99) 767 0 R (theorem.3.1) 797 0 R (theorem.3.10) 821 0 R (theorem.3.11) 828 0 R (theorem.3.12) 829 0 R (theorem.3.13) 832 0 R] +/Limits [(theorem.2.99) (theorem.3.13)] >> endobj 1072 0 obj << -/Names [(theorem.3.64) 912 0 R (theorem.3.7) 805 0 R (theorem.3.8) 811 0 R (theorem.3.9) 814 0 R] -/Limits [(theorem.3.64) (theorem.3.9)] +/Names [(theorem.3.14) 833 0 R (theorem.3.15) 834 0 R (theorem.3.16) 835 0 R (theorem.3.17) 836 0 R (theorem.3.18) 837 0 R (theorem.3.19) 838 0 R] +/Limits [(theorem.3.14) (theorem.3.19)] >> endobj 1073 0 obj << -/Kids [1004 0 R 1005 0 R 1006 0 R 1007 0 R 1008 0 R 1009 0 R] -/Limits [(Doc-Start) (Item.33)] +/Names [(theorem.3.2) 798 0 R (theorem.3.20) 839 0 R (theorem.3.21) 840 0 R (theorem.3.22) 853 0 R (theorem.3.23) 858 0 R (theorem.3.24) 859 0 R] +/Limits [(theorem.3.2) (theorem.3.24)] >> endobj 1074 0 obj << -/Kids [1010 0 R 1011 0 R 1012 0 R 1013 0 R 1014 0 R 1015 0 R] -/Limits [(Item.34) (Item.7)] +/Names [(theorem.3.25) 860 0 R (theorem.3.26) 861 0 R (theorem.3.27) 862 0 R (theorem.3.28) 863 0 R (theorem.3.29) 864 0 R (theorem.3.3) 802 0 R] +/Limits [(theorem.3.25) (theorem.3.3)] >> endobj 1075 0 obj << -/Kids [1016 0 R 1017 0 R 1018 0 R 1019 0 R 1020 0 R 1021 0 R] -/Limits [(Item.8) (page.30)] +/Names [(theorem.3.30) 869 0 R (theorem.3.31) 870 0 R (theorem.3.32) 871 0 R (theorem.3.33) 872 0 R (theorem.3.34) 873 0 R (theorem.3.35) 874 0 R] +/Limits [(theorem.3.30) (theorem.3.35)] >> endobj 1076 0 obj << -/Kids [1022 0 R 1023 0 R 1024 0 R 1025 0 R 1026 0 R 1027 0 R] -/Limits [(page.31) (section.3.5)] +/Names [(theorem.3.36) 877 0 R (theorem.3.37) 880 0 R (theorem.3.38) 881 0 R (theorem.3.39) 882 0 R (theorem.3.4) 808 0 R (theorem.3.40) 883 0 R] +/Limits [(theorem.3.36) (theorem.3.40)] >> endobj 1077 0 obj << -/Kids [1028 0 R 1029 0 R 1030 0 R 1031 0 R 1032 0 R 1033 0 R] -/Limits [(section.3.6) (subsection.3.1.2)] +/Names [(theorem.3.41) 884 0 R (theorem.3.42) 885 0 R (theorem.3.43) 886 0 R (theorem.3.44) 887 0 R (theorem.3.45) 894 0 R (theorem.3.46) 895 0 R] +/Limits [(theorem.3.41) (theorem.3.46)] >> endobj 1078 0 obj << -/Kids [1034 0 R 1035 0 R 1036 0 R 1037 0 R 1038 0 R 1039 0 R] -/Limits [(subsection.3.1.3) (theorem.1.3)] +/Names [(theorem.3.47) 896 0 R (theorem.3.48) 897 0 R (theorem.3.49) 898 0 R (theorem.3.5) 809 0 R (theorem.3.50) 899 0 R (theorem.3.51) 900 0 R] +/Limits [(theorem.3.47) (theorem.3.51)] >> endobj 1079 0 obj << -/Kids [1040 0 R 1041 0 R 1042 0 R 1043 0 R 1044 0 R 1045 0 R] -/Limits [(theorem.1.30) (theorem.2.11)] +/Names [(theorem.3.52) 901 0 R (theorem.3.53) 902 0 R (theorem.3.54) 903 0 R (theorem.3.55) 904 0 R (theorem.3.56) 905 0 R (theorem.3.57) 906 0 R] +/Limits [(theorem.3.52) (theorem.3.57)] >> endobj 1080 0 obj << -/Kids [1046 0 R 1047 0 R 1048 0 R 1049 0 R 1050 0 R 1051 0 R] -/Limits [(theorem.2.12) (theorem.2.44)] +/Names [(theorem.3.58) 907 0 R (theorem.3.59) 908 0 R (theorem.3.6) 810 0 R (theorem.3.60) 909 0 R (theorem.3.61) 916 0 R (theorem.3.62) 917 0 R] +/Limits [(theorem.3.58) (theorem.3.62)] >> endobj 1081 0 obj << -/Kids [1052 0 R 1053 0 R 1054 0 R 1055 0 R 1056 0 R 1057 0 R] -/Limits [(theorem.2.45) (theorem.2.77)] +/Names [(theorem.3.63) 918 0 R (theorem.3.64) 919 0 R (theorem.3.65) 923 0 R (theorem.3.66) 924 0 R (theorem.3.7) 811 0 R (theorem.3.8) 817 0 R] +/Limits [(theorem.3.63) (theorem.3.8)] >> endobj 1082 0 obj << -/Kids [1058 0 R 1059 0 R 1060 0 R 1061 0 R 1062 0 R 1063 0 R] -/Limits [(theorem.2.78) (theorem.3.2)] +/Names [(theorem.3.9) 820 0 R] +/Limits [(theorem.3.9) (theorem.3.9)] >> endobj 1083 0 obj << -/Kids [1064 0 R 1065 0 R 1066 0 R 1067 0 R 1068 0 R 1069 0 R] -/Limits [(theorem.3.20) (theorem.3.52)] +/Kids [1013 0 R 1014 0 R 1015 0 R 1016 0 R 1017 0 R 1018 0 R] +/Limits [(Doc-Start) (Item.33)] >> endobj 1084 0 obj << -/Kids [1070 0 R 1071 0 R 1072 0 R] -/Limits [(theorem.3.53) (theorem.3.9)] +/Kids [1019 0 R 1020 0 R 1021 0 R 1022 0 R 1023 0 R 1024 0 R] +/Limits [(Item.34) (Item.7)] >> endobj 1085 0 obj << -/Kids [1073 0 R 1074 0 R 1075 0 R 1076 0 R 1077 0 R 1078 0 R] -/Limits [(Doc-Start) (theorem.1.3)] +/Kids [1025 0 R 1026 0 R 1027 0 R 1028 0 R 1029 0 R 1030 0 R] +/Limits [(Item.8) (page.30)] >> endobj 1086 0 obj << -/Kids [1079 0 R 1080 0 R 1081 0 R 1082 0 R 1083 0 R 1084 0 R] -/Limits [(theorem.1.30) (theorem.3.9)] +/Kids [1031 0 R 1032 0 R 1033 0 R 1034 0 R 1035 0 R 1036 0 R] +/Limits [(page.31) (section.3.5)] >> endobj 1087 0 obj << -/Kids [1085 0 R 1086 0 R] -/Limits [(Doc-Start) (theorem.3.9)] +/Kids [1037 0 R 1038 0 R 1039 0 R 1040 0 R 1041 0 R 1042 0 R] +/Limits [(section.3.6) (subsection.3.1.1)] >> endobj 1088 0 obj << -/Dests 1087 0 R +/Kids [1043 0 R 1044 0 R 1045 0 R 1046 0 R 1047 0 R 1048 0 R] +/Limits [(subsection.3.1.2) (theorem.1.29)] >> endobj 1089 0 obj << +/Kids [1049 0 R 1050 0 R 1051 0 R 1052 0 R 1053 0 R 1054 0 R] +/Limits [(theorem.1.3) (theorem.2.108)] +>> +endobj +1090 0 obj +<< +/Kids [1055 0 R 1056 0 R 1057 0 R 1058 0 R 1059 0 R 1060 0 R] +/Limits [(theorem.2.11) (theorem.2.43)] +>> +endobj +1091 0 obj +<< +/Kids [1061 0 R 1062 0 R 1063 0 R 1064 0 R 1065 0 R 1066 0 R] +/Limits [(theorem.2.44) (theorem.2.76)] +>> +endobj +1092 0 obj +<< +/Kids [1067 0 R 1068 0 R 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00000 n +0000425620 00000 n +0000425706 00000 n +0000425746 00000 n +0000425918 00000 n trailer -<< /Size 1091 -/Root 1089 0 R -/Info 1090 0 R -/ID [<1BDC533C42D8B1020F699175D5EDC7C2> <1BDC533C42D8B1020F699175D5EDC7C2>] >> +<< /Size 1101 +/Root 1099 0 R +/Info 1100 0 R +/ID [<3AFD012FA29F523F54560632BD2257DD> <3AFD012FA29F523F54560632BD2257DD>] >> startxref -421151 +426193 %%EOF diff --git a/Secondo anno/Elementi di probabilità e statistica/sections/0-notazioni.tex b/Secondo anno/Elementi di probabilità e statistica/sections/0-notazioni.tex index fcb858a..7033215 100644 --- a/Secondo anno/Elementi di probabilità e statistica/sections/0-notazioni.tex +++ b/Secondo anno/Elementi di probabilità e statistica/sections/0-notazioni.tex @@ -5,7 +5,7 @@ \begin{multicols*}{2} \section*{Algebra lineare} - \addcontentsline{toc}{section}{Teoria della misura} + \addcontentsline{toc}{section}{Algebra lineare} \begin{itemize} \item $q_\varphi$ -- dato uno spazio vettoriale $V$ equipaggiato con un @@ -157,6 +157,9 @@ \item $\pi$-sistema -- insieme $I \subseteq \FF$, $I \neq \emptyset$ con $(\Omega, \FF)$ spazio misurabile, $\sigma(I) = \FF$ e $I$ chiuso per intersezioni. \item $\mu$ -- misura su uno spazio misurabile. \item $m$ -- misura di Lebesgue sullo spazio misurabile $(\RR, \BB(\RR))$. È tale per cui $m([a, b]) = b-a$ per $b > a$. + \item $m$ -- misura di Lebesgue sullo spazio misurabile $\left(\RR^d, \BB\left(\RR^d\right)\right)$ con $d \geq 1$. È tale per cui + $m\left([a_1, b_1] \times \cdots \times [a_d, b_d]\right) = (b_1 - a_1) \cdots (b_d - a_d)$ con $a_i$, $b_i \in \RR$ e + $b_i > a_i$ per $1 \leq i \leq d$. Non si distingue generalmente la notazione dal caso unidimensionale. \item $P$ -- misura di probabilità su uno spazio misurabile. \item $(\Omega, \FF, P)$ -- spazio di probabilità. \item \qc -- quasi certo/quasi certamente. diff --git a/Secondo anno/Elementi di probabilità e statistica/sections/0-prerequisiti.tex b/Secondo anno/Elementi di probabilità e statistica/sections/0-prerequisiti.tex index 5b36d8b..01500f1 100644 --- a/Secondo anno/Elementi di probabilità e statistica/sections/0-prerequisiti.tex +++ b/Secondo anno/Elementi di probabilità e statistica/sections/0-prerequisiti.tex @@ -65,13 +65,17 @@ $\abs{x}$ e $\abs{y}$ ($t^p$ è convessa per $t \geq 0$). \item \textbf{Una funzione crescente ammette un insieme discreto di discontinuità} -- È possibile costruire facilmente una funzione iniettiva da tale insieme a $\QQ$ sfruttando - i limiti sinistri nelle discontinuità. + i limiti sinistri e destri nelle discontinuità. \item \textbf{Lemma di Dynkin, versione probabilistica} -- Se due misure di probabilità $P$ e $Q$ su $(\Omega, \FF)$ coincidono su un $\pi$-sistema di $\FF$ contenente $\Omega$, allora $P \equiv Q$. \item \textbf{Esistenza e unicità della misura di Lebesgue} -- Esiste ed è unica la misura $m$ su $(\RR, \BB(\RR))$ tale per cui $m([a, b]) = b-a$. Segue dalla versione più generale del lemma di Dynkin. + \item \textbf{Esistenza e unicità della misura di Lebesgue $d$-dimensionale} -- Esiste ed + è unica la misura $m$ su $\left(\RR^d, \BB\left(\RR^d\right)\right)$ tale per cui + $m\left([a_1, b_1] \times \cdots \times [a_d, b_d]\right) = (b_1 - a_1) \cdots (b_d - a_d)$ con $a_i$, $b_i \in \RR$ e + $b_i > a_i$ per $1 \leq i \leq d$. \item \textbf{Lemma di Fatou} -- Sia $(X, \FF)$ uno spazio misurabile e sia $(f_i : X \to \RR)_{i \in \NN}$ una successione di funzioni misurabili rispetto a $(\RR, m)$ con $f_i \geq 0$ e con diff --git a/Secondo anno/Elementi di probabilità e statistica/sections/2-probabilità-discreta.tex b/Secondo anno/Elementi di probabilità e statistica/sections/2-probabilità-discreta.tex index 894aa40..26fe79c 100644 --- a/Secondo anno/Elementi di probabilità e statistica/sections/2-probabilità-discreta.tex +++ b/Secondo anno/Elementi di probabilità e statistica/sections/2-probabilità-discreta.tex @@ -653,6 +653,7 @@ aspettarsi il seguente: \end{lemma} \subsection{Momenti (assoluti) \texorpdfstring{$n$}{n}-esimi} +\label{sec:momenti_assoluti} \begin{definition}[Momento $n$-esimo assoluto] Data $X$ v.a.~reale e $n \in \RR^+$, definiamo il diff --git a/Secondo anno/Elementi di probabilità e statistica/sections/3-probabilità-reale.tex b/Secondo anno/Elementi di probabilità e statistica/sections/3-probabilità-reale.tex index 25a52b4..1d28645 100644 --- a/Secondo anno/Elementi di probabilità e statistica/sections/3-probabilità-reale.tex +++ b/Secondo anno/Elementi di probabilità e statistica/sections/3-probabilità-reale.tex @@ -260,7 +260,9 @@ piccola di $\PP(\RR)$, la $\sigma$-algebra dei boreliani, che ci permette di esc $F$ come funzione di ripartizione. \smallskip - L'unicità segue dal lemma di Dynkin. + L'unicità segue dal lemma di Dynkin. Per l'esistenza è utile considerare + che l'insieme delle discontinuità di $F$ è discreto dacché + $F$ è crescente. \end{proposition} \begin{remark} @@ -717,4 +719,33 @@ che $\mu$ assuma valori finiti per le funzioni semplici). \hyperref[th:convergenza_monotona]{Teorema di convergenza monotona}. \end{proposition} +\begin{remark} + In particolare, per $X$ v.a.~assolutamente continua vale che: + \[ + \EE[X] = \int_\RR x f(x) \dx. + \] +\end{remark} + +\section{Momenti e disuguaglianze, (co)varianza, dev.~standard, mediana e moda} + +Tutti le disuguaglianze sul valore atteso (e.g.~Markov) e tutti i risultati +riguardanti i momenti (assoluti e non), la covarianza, la varianza, la +deviazione standard, la mediana e la moda passano direttamente al +caso reale a partire dalle proprietà del funzionale lineare +$\EE[\cdot]$. Si rimanda dunque alla \hyperref[sec:momenti_assoluti]{\textit{Parte 2}}. + +\begin{proposition} + Sia $X$ v.a.~reale e siano $A$ e $B$ tali per cui: + \[ + A = \left\{t \in \RR \mid F(t) < \frac{1}{2} \right\}, \quad B = \left\{t \in \RR \mid F(t) > \frac{1}{2} \right\}. + \] + Allora $m$ è una mediana se e solo se $m \in [\underline{m}, \overline{m}]$, dove $\underline{m} = \sup \, A$ e + $\overline{m} = \inf \, B$. \smallskip + + + Infatti ogni mediana $m$ è maggiorante di $A$ e minorante di $B$ per la monotonia di + $F$ (e dunque $m \in [\underline{m}, \overline{m}]$). Si verifica poi che ogni elemento + di tale intervallo è mediana. +\end{proposition} + \end{multicols*} \ No newline at end of file