From e71b07726dfa724e0722ce159fe9e3de541067ef Mon Sep 17 00:00:00 2001 From: Hearot Date: Sat, 20 Apr 2024 14:53:52 +0200 Subject: [PATCH] feat(eps): inizia la Parte III --- .../main.pdf | 5216 +++++++++-------- .../main.tex | 3 +- .../preamble.tex | 2 +- .../sections/0-prerequisiti.tex | 15 + .../sections/1-spazi-in-generale.tex | 2 +- .../sections/3-probabilità-reale.tex | 78 + 6 files changed, 2979 insertions(+), 2337 deletions(-) create mode 100644 Secondo anno/Elementi di probabilità e statistica/sections/3-probabilità-reale.tex diff --git a/Secondo anno/Elementi di probabilità e statistica/main.pdf b/Secondo anno/Elementi di probabilità e statistica/main.pdf index 133b9c8..3c2f230 100644 --- a/Secondo anno/Elementi di probabilità e statistica/main.pdf +++ b/Secondo anno/Elementi di probabilità e statistica/main.pdf @@ -13,1065 +13,1209 @@ endobj (\376\377\000P\000r\000e\000r\000e\000q\000u\000i\000s\000i\000t\000i\000\040\000m\000a\000t\000e\000m\000a\000t\000i\000c\000i) endobj 12 0 obj -<< /S /GoTo /D (chapter*.13) >> +<< /S /GoTo /D (section*.9) >> endobj 15 0 obj -(\376\377\000L\000i\000s\000t\000a\000\040\000d\000e\000l\000l\000e\000\040\000i\000d\000e\000n\000t\000i\000t\000\340\000\040\000s\000u\000l\000l\000e\000\040\000s\000o\000m\000m\000a\000t\000o\000r\000i\000e) +(\376\377\000A\000l\000g\000e\000b\000r\000a\000\040\000l\000i\000n\000e\000a\000r\000e) endobj 16 0 obj -<< /S /GoTo /D (chapter.1) >> +<< /S /GoTo /D (section*.10) >> endobj 19 0 obj -(\376\377\000S\000p\000a\000z\000i\000\040\000d\000i\000\040\000p\000r\000o\000b\000a\000b\000i\000l\000i\000t\000\340\000\040\000i\000n\000\040\000g\000e\000n\000e\000r\000a\000l\000e) +(\376\377\000A\000n\000a\000l\000i\000s\000i\000\040\000m\000a\000t\000e\000m\000a\000t\000i\000c\000a) endobj 20 0 obj -<< /S /GoTo /D (section.1.1) >> +<< /S /GoTo /D (section*.11) >> endobj 23 0 obj -(\376\377\000D\000e\000f\000i\000n\000i\000z\000i\000o\000n\000i\000\040\000p\000r\000e\000l\000i\000m\000i\000n\000a\000r\000i) +(\376\377\000C\000o\000m\000b\000i\000n\000a\000t\000o\000r\000i\000a) endobj 24 0 obj -<< /S /GoTo /D (subsection.1.1.1) >> +<< /S /GoTo /D (section*.12) >> endobj 27 0 obj -(\376\377\000E\000s\000p\000e\000r\000i\000m\000e\000n\000t\000o\000\040\000a\000l\000e\000a\000t\000o\000r\000i\000o\000,\000\040\000s\000p\000a\000z\000i\000\040\000c\000a\000m\000p\000i\000o\000n\000a\000r\000i) +(\376\377\000T\000e\000o\000r\000i\000a\000\040\000d\000e\000g\000l\000i\000\040\000i\000n\000s\000i\000e\000m\000i) endobj 28 0 obj -<< /S /GoTo /D (subsection.1.1.2) >> +<< /S /GoTo /D (section*.13) >> endobj 31 0 obj -(\376\377\003\303\000-\000a\000l\000g\000e\000b\000r\000e\000\040\000e\000\040\000s\000p\000a\000z\000i\000\040\000m\000i\000s\000u\000r\000a\000b\000i\000l\000i) +(\376\377\000T\000e\000o\000r\000i\000a\000\040\000d\000e\000l\000l\000a\000\040\000m\000i\000s\000u\000r\000a) endobj 32 0 obj -<< /S /GoTo /D (subsection.1.1.3) >> +<< /S /GoTo /D (chapter*.14) >> endobj 35 0 obj -(\376\377\000I\000n\000s\000i\000e\000m\000i\000\040\000d\000i\000s\000c\000r\000e\000t\000i\000\040\000e\000\040\003\303\000-\000a\000l\000g\000e\000b\000r\000a\000\040\000n\000a\000t\000u\000r\000a\000l\000e) +(\376\377\000L\000i\000s\000t\000a\000\040\000d\000e\000l\000l\000e\000\040\000i\000d\000e\000n\000t\000i\000t\000\340\000\040\000s\000u\000l\000l\000e\000\040\000s\000o\000m\000m\000a\000t\000o\000r\000i\000e) endobj 36 0 obj -<< /S /GoTo /D (subsection.1.1.4) >> +<< /S /GoTo /D (chapter.1) >> endobj 39 0 obj -(\376\377\000P\000r\000o\000p\000r\000i\000e\000t\000\340\000\040\000d\000i\000\040\000u\000n\000a\000\040\003\303\000-\000a\000l\000g\000e\000b\000r\000a\000\040\000e\000\040\003\303\000-\000a\000l\000g\000e\000b\000r\000a\000\040\000g\000e\000n\000e\000r\000a\000t\000a) +(\376\377\000S\000p\000a\000z\000i\000\040\000d\000i\000\040\000p\000r\000o\000b\000a\000b\000i\000l\000i\000t\000\340\000\040\000i\000n\000\040\000g\000e\000n\000e\000r\000a\000l\000e) endobj 40 0 obj -<< /S /GoTo /D (section.1.2) >> +<< /S /GoTo /D (section.1.1) >> endobj 43 0 obj -(\376\377\000C\000o\000r\000r\000i\000s\000p\000o\000n\000d\000e\000n\000z\000e\000\040\000l\000o\000g\000i\000c\000h\000e\000\040\000e\000\040\000r\000e\000l\000a\000z\000i\000o\000n\000a\000l\000i\000\040\000t\000r\000a\000\040\000e\000v\000e\000n\000t\000i) +(\376\377\000D\000e\000f\000i\000n\000i\000z\000i\000o\000n\000i\000\040\000p\000r\000e\000l\000i\000m\000i\000n\000a\000r\000i) endobj 44 0 obj -<< /S /GoTo /D (section.1.3) >> +<< /S /GoTo /D (subsection.1.1.1) >> endobj 47 0 obj -(\376\377\000M\000i\000s\000u\000r\000e\000\040\000d\000i\000\040\000p\000r\000o\000b\000a\000b\000i\000l\000i\000t\000\340) +(\376\377\000E\000s\000p\000e\000r\000i\000m\000e\000n\000t\000o\000\040\000a\000l\000e\000a\000t\000o\000r\000i\000o\000,\000\040\000s\000p\000a\000z\000i\000\040\000c\000a\000m\000p\000i\000o\000n\000a\000r\000i) endobj 48 0 obj -<< /S /GoTo /D (subsection.1.3.1) >> +<< /S /GoTo /D (subsection.1.1.2) >> endobj 51 0 obj -(\376\377\000L\000a\000\040\000p\000r\000o\000b\000a\000b\000i\000l\000i\000t\000\340\000\040\000P\000\040\000s\000u\000\040\003\251\000\040\000e\000\040\000s\000p\000a\000z\000i\000\040\000d\000i\000\040\000p\000r\000o\000b\000a\000b\000i\000l\000i\000t\000\340) +(\376\377\003\303\000-\000a\000l\000g\000e\000b\000r\000e\000\040\000e\000\040\000s\000p\000a\000z\000i\000\040\000m\000i\000s\000u\000r\000a\000b\000i\000l\000i) endobj 52 0 obj -<< /S /GoTo /D (subsection.1.3.2) >> +<< /S /GoTo /D (subsection.1.1.3) >> endobj 55 0 obj -(\376\377\000P\000r\000o\000p\000r\000i\000e\000t\000\340\000\040\000d\000e\000l\000l\000a\000\040\000p\000r\000o\000b\000a\000b\000i\000l\000i\000t\000\340\000\040\000P) +(\376\377\000I\000n\000s\000i\000e\000m\000i\000\040\000d\000i\000s\000c\000r\000e\000t\000i\000\040\000e\000\040\003\303\000-\000a\000l\000g\000e\000b\000r\000a\000\040\000n\000a\000t\000u\000r\000a\000l\000e) endobj 56 0 obj -<< /S /GoTo /D (subsection.1.3.3) >> +<< /S /GoTo /D (subsection.1.1.4) >> endobj 59 0 obj -(\376\377\000E\000v\000e\000n\000t\000i\000\040\000i\000n\000c\000o\000m\000p\000a\000t\000i\000b\000i\000l\000i\000,\000\040\000q\000u\000a\000s\000i\000\040\000c\000e\000r\000t\000i\000\040\000e\000\040\000t\000r\000a\000s\000c\000u\000r\000a\000b\000i\000l\000i\000,\000\040\000p\000r\000o\000p\000r\000i\000e\000t\000\340\000\040\000c\000h\000e\000\040\000a\000c\000c\000a\000d\000o\000n\000o\000\040\000q\000.\000c\000.) +(\376\377\000P\000r\000o\000p\000r\000i\000e\000t\000\340\000\040\000d\000i\000\040\000u\000n\000a\000\040\003\303\000-\000a\000l\000g\000e\000b\000r\000a\000\040\000e\000\040\003\303\000-\000a\000l\000g\000e\000b\000r\000a\000\040\000g\000e\000n\000e\000r\000a\000t\000a) endobj 60 0 obj -<< /S /GoTo /D (section.1.4) >> +<< /S /GoTo /D (section.1.2) >> endobj 63 0 obj -(\376\377\000P\000r\000o\000b\000a\000b\000i\000l\000i\000t\000\340\000\040\000c\000o\000n\000d\000i\000z\000i\000o\000n\000a\000t\000a) +(\376\377\000C\000o\000r\000r\000i\000s\000p\000o\000n\000d\000e\000n\000z\000e\000\040\000l\000o\000g\000i\000c\000h\000e\000\040\000e\000\040\000r\000e\000l\000a\000z\000i\000o\000n\000a\000l\000i\000\040\000t\000r\000a\000\040\000e\000v\000e\000n\000t\000i) endobj 64 0 obj -<< /S /GoTo /D (subsection.1.4.1) >> +<< /S /GoTo /D (section.1.3) >> endobj 67 0 obj -(\376\377\000D\000e\000f\000i\000n\000i\000z\000i\000o\000n\000e\000\040\000d\000i\000\040\000P\000\050\040\042\000|\000B\000\051) +(\376\377\000M\000i\000s\000u\000r\000e\000\040\000d\000i\000\040\000p\000r\000o\000b\000a\000b\000i\000l\000i\000t\000\340) endobj 68 0 obj -<< /S /GoTo /D (subsection.1.4.2) >> +<< /S /GoTo /D (subsection.1.3.1) >> endobj 71 0 obj -(\376\377\000R\000e\000g\000o\000l\000a\000\040\000d\000e\000l\000l\000a\000\040\000c\000a\000t\000e\000n\000a\000,\000\040\000f\000o\000r\000m\000u\000l\000a\000\040\000d\000e\000l\000l\000e\000\040\000p\000r\000o\000b\000a\000b\000i\000l\000i\000t\000\340\000\040\000t\000o\000t\000a\000l\000i\000\040\000e\000\040\000T\000e\000o\000r\000e\000m\000a\000\040\000d\000i\000\040\000B\000a\000y\000e\000s) +(\376\377\000L\000a\000\040\000p\000r\000o\000b\000a\000b\000i\000l\000i\000t\000\340\000\040\000P\000\040\000s\000u\000\040\003\251\000\040\000e\000\040\000s\000p\000a\000z\000i\000\040\000d\000i\000\040\000p\000r\000o\000b\000a\000b\000i\000l\000i\000t\000\340) endobj 72 0 obj -<< /S /GoTo /D (subsection.1.4.3) >> +<< /S /GoTo /D (subsection.1.3.2) >> endobj 75 0 obj -(\376\377\000R\000a\000p\000p\000o\000r\000t\000o\000\040\000d\000i\000\040\000i\000n\000f\000l\000u\000e\000n\000z\000a\000,\000\040\000c\000o\000r\000r\000e\000l\000a\000z\000i\000o\000n\000e\000\040\000p\000o\000s\000i\000t\000i\000v\000a\000\040\000e\000\040\000n\000e\000g\000a\000t\000i\000v\000a) +(\376\377\000P\000r\000o\000p\000r\000i\000e\000t\000\340\000\040\000d\000e\000l\000l\000a\000\040\000p\000r\000o\000b\000a\000b\000i\000l\000i\000t\000\340\000\040\000P) endobj 76 0 obj -<< /S /GoTo /D (section.1.5) >> +<< /S /GoTo /D (subsection.1.3.3) >> endobj 79 0 obj -(\376\377\000I\000n\000d\000i\000p\000e\000n\000d\000e\000n\000z\000a\000\040\000s\000t\000o\000c\000a\000s\000t\000i\000c\000a\000\040\000t\000r\000a\000\040\000e\000v\000e\000n\000t\000i) +(\376\377\000E\000v\000e\000n\000t\000i\000\040\000i\000n\000c\000o\000m\000p\000a\000t\000i\000b\000i\000l\000i\000,\000\040\000q\000u\000a\000s\000i\000\040\000c\000e\000r\000t\000i\000\040\000e\000\040\000t\000r\000a\000s\000c\000u\000r\000a\000b\000i\000l\000i\000,\000\040\000p\000r\000o\000p\000r\000i\000e\000t\000\340\000\040\000c\000h\000e\000\040\000a\000c\000c\000a\000d\000o\000n\000o\000\040\000q\000.\000c\000.) endobj 80 0 obj -<< /S /GoTo /D (chapter.2) >> +<< /S /GoTo /D (section.1.4) >> endobj 83 0 obj -(\376\377\000P\000r\000o\000b\000a\000b\000i\000l\000i\000t\000\340\000\040\000d\000i\000s\000c\000r\000e\000t\000a) +(\376\377\000P\000r\000o\000b\000a\000b\000i\000l\000i\000t\000\340\000\040\000c\000o\000n\000d\000i\000z\000i\000o\000n\000a\000t\000a) endobj 84 0 obj -<< /S /GoTo /D (section.2.1) >> +<< /S /GoTo /D (subsection.1.4.1) >> endobj 87 0 obj -(\376\377\000F\000u\000n\000z\000i\000o\000n\000e\000\040\000d\000i\000\040\000d\000e\000n\000s\000i\000t\000\340\000\040\000d\000i\000s\000c\000r\000e\000t\000a) +(\376\377\000D\000e\000f\000i\000n\000i\000z\000i\000o\000n\000e\000\040\000d\000i\000\040\000P\000\050\040\042\000|\000B\000\051) endobj 88 0 obj -<< /S /GoTo /D (subsection.2.1.1) >> +<< /S /GoTo /D (subsection.1.4.2) >> endobj 91 0 obj -(\376\377\000D\000e\000f\000i\000n\000i\000z\000i\000o\000n\000e\000\040\000p\000e\000r\000\040\000i\000l\000\040\000c\000a\000s\000o\000\040\000d\000i\000s\000c\000r\000e\000t\000o) +(\376\377\000R\000e\000g\000o\000l\000a\000\040\000d\000e\000l\000l\000a\000\040\000c\000a\000t\000e\000n\000a\000,\000\040\000f\000o\000r\000m\000u\000l\000a\000\040\000d\000e\000l\000l\000e\000\040\000p\000r\000o\000b\000a\000b\000i\000l\000i\000t\000\340\000\040\000t\000o\000t\000a\000l\000i\000\040\000e\000\040\000T\000e\000o\000r\000e\000m\000a\000\040\000d\000i\000\040\000B\000a\000y\000e\000s) endobj 92 0 obj -<< /S /GoTo /D (subsection.2.1.2) >> +<< /S /GoTo /D (subsection.1.4.3) >> endobj 95 0 obj -(\376\377\000R\000a\000n\000g\000e\000\040\000d\000i\000\040\000u\000n\000a\000\040\000p\000r\000o\000b\000a\000b\000i\000l\000i\000t\000\340\000\040\000d\000i\000s\000c\000r\000e\000t\000a\000\040\000e\000\040\000r\000e\000s\000t\000r\000i\000z\000i\000o\000n\000e) +(\376\377\000R\000a\000p\000p\000o\000r\000t\000o\000\040\000d\000i\000\040\000i\000n\000f\000l\000u\000e\000n\000z\000a\000,\000\040\000c\000o\000r\000r\000e\000l\000a\000z\000i\000o\000n\000e\000\040\000p\000o\000s\000i\000t\000i\000v\000a\000\040\000e\000\040\000n\000e\000g\000a\000t\000i\000v\000a) endobj 96 0 obj -<< /S /GoTo /D (subsection.2.1.3) >> +<< /S /GoTo /D (section.1.5) >> endobj 99 0 obj -(\376\377\000M\000i\000s\000u\000r\000e\000\040\000d\000i\000\040\000p\000r\000o\000b\000a\000b\000i\000l\000i\000t\000\340\000\040\000d\000i\000s\000c\000r\000e\000t\000e\000\040\000s\000u\000\040\000s\000p\000a\000z\000i\000\040\000c\000a\000m\000p\000i\000o\000n\000a\000r\000i\000\040\000n\000o\000n\000\040\000d\000i\000s\000c\000r\000e\000t\000i\000\040\000e\000\040\000d\000i\000s\000c\000r\000e\000t\000i\000z\000z\000a\000z\000i\000o\000n\000e) +(\376\377\000I\000n\000d\000i\000p\000e\000n\000d\000e\000n\000z\000a\000\040\000s\000t\000o\000c\000a\000s\000t\000i\000c\000a\000\040\000t\000r\000a\000\040\000e\000v\000e\000n\000t\000i) endobj 100 0 obj -<< /S /GoTo /D (section.2.2) >> +<< /S /GoTo /D (chapter.2) >> endobj 103 0 obj -(\376\377\000V\000a\000r\000i\000a\000b\000i\000l\000i\000\040\000a\000l\000e\000a\000t\000o\000r\000i\000e\000\040\000d\000i\000s\000c\000r\000e\000t\000e) +(\376\377\000P\000r\000o\000b\000a\000b\000i\000l\000i\000t\000\340\000\040\000d\000i\000s\000c\000r\000e\000t\000a) endobj 104 0 obj -<< /S /GoTo /D (subsection.2.2.1) >> +<< /S /GoTo /D (section.2.1) >> endobj 107 0 obj -(\376\377\000D\000e\000f\000i\000n\000i\000z\000i\000o\000n\000e\000\040\000d\000i\000\040\000v\000.\000a\000.\000\040\000d\000i\000s\000c\000r\000e\000t\000a\000\040\000e\000\040\000c\000o\000m\000p\000o\000s\000i\000z\000i\000o\000n\000e) +(\376\377\000F\000u\000n\000z\000i\000o\000n\000e\000\040\000d\000i\000\040\000d\000e\000n\000s\000i\000t\000\340\000\040\000d\000i\000s\000c\000r\000e\000t\000a) endobj 108 0 obj -<< /S /GoTo /D (subsection.2.2.2) >> +<< /S /GoTo /D (subsection.2.1.1) >> endobj 111 0 obj -(\376\377\000L\000e\000g\000g\000e\000\040\000d\000i\000\040\000u\000n\000a\000\040\000v\000.\000a\000.\000\040\000X) +(\376\377\000D\000e\000f\000i\000n\000i\000z\000i\000o\000n\000e\000\040\000p\000e\000r\000\040\000i\000l\000\040\000c\000a\000s\000o\000\040\000d\000i\000s\000c\000r\000e\000t\000o) endobj 112 0 obj -<< /S /GoTo /D (subsection.2.2.3) >> +<< /S /GoTo /D (subsection.2.1.2) >> endobj 115 0 obj -(\376\377\000U\000g\000u\000a\000g\000l\000i\000a\000n\000z\000a\000\040\000q\000.\000c\000.\000,\000\040\000m\000e\000d\000e\000s\000i\000m\000a\000\040\000l\000e\000g\000g\000e\000\040\000e\000\040\000s\000t\000a\000b\000i\000l\000i\000t\000\340\000\040\000p\000e\000r\000\040\000c\000o\000m\000p\000o\000s\000i\000z\000i\000o\000n\000e) +(\376\377\000R\000a\000n\000g\000e\000\040\000d\000i\000\040\000u\000n\000a\000\040\000p\000r\000o\000b\000a\000b\000i\000l\000i\000t\000\340\000\040\000d\000i\000s\000c\000r\000e\000t\000a\000\040\000e\000\040\000r\000e\000s\000t\000r\000i\000z\000i\000o\000n\000e) endobj 116 0 obj -<< /S /GoTo /D (subsection.2.2.4) >> +<< /S /GoTo /D (subsection.2.1.3) >> endobj 119 0 obj -(\376\377\000V\000a\000r\000i\000a\000b\000i\000l\000e\000\040\000a\000l\000e\000a\000t\000o\000r\000i\000a\000\040\000m\000u\000l\000t\000i\000v\000a\000r\000i\000a\000t\000a\000,\000\040\000l\000e\000g\000g\000i\000\040\000c\000o\000n\000g\000i\000u\000n\000t\000e\000\040\000e\000\040\000m\000a\000r\000g\000i\000n\000a\000l\000i) +(\376\377\000M\000i\000s\000u\000r\000e\000\040\000d\000i\000\040\000p\000r\000o\000b\000a\000b\000i\000l\000i\000t\000\340\000\040\000d\000i\000s\000c\000r\000e\000t\000e\000\040\000s\000u\000\040\000s\000p\000a\000z\000i\000\040\000c\000a\000m\000p\000i\000o\000n\000a\000r\000i\000\040\000n\000o\000n\000\040\000d\000i\000s\000c\000r\000e\000t\000i\000\040\000e\000\040\000d\000i\000s\000c\000r\000e\000t\000i\000z\000z\000a\000z\000i\000o\000n\000e) endobj 120 0 obj -<< /S /GoTo /D (subsection.2.2.5) >> +<< /S /GoTo /D (section.2.2) >> endobj 123 0 obj -(\376\377\000I\000n\000d\000i\000p\000e\000n\000d\000e\000n\000z\000a\000\040\000d\000i\000\040\000v\000a\000r\000i\000a\000b\000i\000l\000i\000\040\000a\000l\000e\000a\000t\000o\000r\000i\000e\000\040\000d\000i\000s\000c\000r\000e\000t\000e\000\040\000e\000\040\000s\000t\000a\000b\000i\000l\000i\000t\000\340\000\040\000p\000e\000r\000\040\000c\000o\000n\000g\000i\000u\000n\000z\000i\000o\000n\000e\000\040\000e\000\040\000c\000o\000m\000p\000o\000s\000i\000z\000i\000o\000n\000e) +(\376\377\000V\000a\000r\000i\000a\000b\000i\000l\000i\000\040\000a\000l\000e\000a\000t\000o\000r\000i\000e\000\040\000d\000i\000s\000c\000r\000e\000t\000e) endobj 124 0 obj -<< /S /GoTo /D (section.2.3) >> +<< /S /GoTo /D (subsection.2.2.1) >> endobj 127 0 obj -(\376\377\000V\000a\000l\000o\000r\000e\000\040\000a\000t\000t\000e\000s\000o\000\040\000e\000\040\000m\000o\000m\000e\000n\000t\000i) +(\376\377\000D\000e\000f\000i\000n\000i\000z\000i\000o\000n\000e\000\040\000d\000i\000\040\000v\000.\000a\000.\000\040\000d\000i\000s\000c\000r\000e\000t\000a\000\040\000e\000\040\000c\000o\000m\000p\000o\000s\000i\000z\000i\000o\000n\000e) endobj 128 0 obj -<< /S /GoTo /D (subsection.2.3.1) >> +<< /S /GoTo /D (subsection.2.2.2) >> endobj 131 0 obj -(\376\377\000V\000a\000l\000o\000r\000e\000\040\000a\000t\000t\000e\000s\000o\000\040\000s\000u\000\040\000v\000.\000a\000.\000\040\000i\000n\000t\000e\000g\000r\000a\000b\000i\000l\000i\000\040\000e\000/\000o\000\040\000n\000o\000n\000\040\000n\000e\000g\000a\000t\000i\000v\000e) +(\376\377\000L\000e\000g\000g\000e\000\040\000d\000i\000\040\000u\000n\000a\000\040\000v\000.\000a\000.\000\040\000X) endobj 132 0 obj -<< /S /GoTo /D (subsection.2.3.2) >> +<< /S /GoTo /D (subsection.2.2.3) >> endobj 135 0 obj -(\376\377\000P\000r\000o\000p\000r\000i\000e\000t\000\340\000\040\000d\000e\000l\000\040\000v\000a\000l\000o\000r\000e\000\040\000a\000t\000t\000e\000s\000o\000\040\000e\000\040\000m\000o\000l\000t\000i\000p\000l\000i\000c\000a\000t\000i\000v\000i\000t\000\340\000\040\000p\000e\000r\000\040\000v\000.\000a\000.\000\040\000i\000n\000d\000i\000p\000e\000n\000d\000e\000n\000t\000i) +(\376\377\000U\000g\000u\000a\000g\000l\000i\000a\000n\000z\000a\000\040\000q\000.\000c\000.\000,\000\040\000m\000e\000d\000e\000s\000i\000m\000a\000\040\000l\000e\000g\000g\000e\000\040\000e\000\040\000s\000t\000a\000b\000i\000l\000i\000t\000\340\000\040\000p\000e\000r\000\040\000c\000o\000m\000p\000o\000s\000i\000z\000i\000o\000n\000e) endobj 136 0 obj -<< /S /GoTo /D (subsection.2.3.3) >> +<< /S /GoTo /D (subsection.2.2.4) >> endobj 139 0 obj -(\376\377\000V\000a\000l\000o\000r\000e\000\040\000a\000t\000t\000e\000s\000o\000\040\000c\000o\000n\000d\000i\000z\000i\000o\000n\000a\000l\000e) +(\376\377\000V\000a\000r\000i\000a\000b\000i\000l\000e\000\040\000a\000l\000e\000a\000t\000o\000r\000i\000a\000\040\000m\000u\000l\000t\000i\000v\000a\000r\000i\000a\000t\000a\000,\000\040\000l\000e\000g\000g\000i\000\040\000c\000o\000n\000g\000i\000u\000n\000t\000e\000\040\000e\000\040\000m\000a\000r\000g\000i\000n\000a\000l\000i) endobj 140 0 obj -<< /S /GoTo /D (subsection.2.3.4) >> +<< /S /GoTo /D (subsection.2.2.5) >> endobj 143 0 obj -(\376\377\000M\000o\000m\000e\000n\000t\000i\000\040\000\050\000a\000s\000s\000o\000l\000u\000t\000i\000\051\000\040\000n\000-\000e\000s\000i\000m\000i) +(\376\377\000I\000n\000d\000i\000p\000e\000n\000d\000e\000n\000z\000a\000\040\000d\000i\000\040\000v\000a\000r\000i\000a\000b\000i\000l\000i\000\040\000a\000l\000e\000a\000t\000o\000r\000i\000e\000\040\000d\000i\000s\000c\000r\000e\000t\000e\000\040\000e\000\040\000s\000t\000a\000b\000i\000l\000i\000t\000\340\000\040\000p\000e\000r\000\040\000c\000o\000n\000g\000i\000u\000n\000z\000i\000o\000n\000e\000\040\000e\000\040\000c\000o\000m\000p\000o\000s\000i\000z\000i\000o\000n\000e) endobj 144 0 obj -<< /S /GoTo /D (subsection.2.3.5) >> +<< /S /GoTo /D (section.2.3) >> endobj 147 0 obj -(\376\377\000D\000i\000s\000u\000g\000u\000a\000g\000l\000i\000a\000n\000z\000a\000\040\000d\000i\000\040\000M\000a\000r\000k\000o\000v\000,\000\040\000d\000i\000\040\000H\000\366\000l\000d\000e\000r\000,\000\040\000d\000i\000\040\000C\000a\000u\000c\000h\000y\000-\000S\000c\000h\000w\000a\000r\000z\000\040\000e\000\040\000d\000i\000\040\000J\000e\000n\000s\000e\000n) +(\376\377\000V\000a\000l\000o\000r\000e\000\040\000a\000t\000t\000e\000s\000o\000\040\000e\000\040\000m\000o\000m\000e\000n\000t\000i) endobj 148 0 obj -<< /S /GoTo /D (section.2.4) >> +<< /S /GoTo /D (subsection.2.3.1) >> endobj 151 0 obj -(\376\377\000A\000l\000t\000r\000i\000\040\000i\000n\000d\000i\000c\000i\000\040\000d\000i\000\040\000c\000e\000n\000t\000r\000a\000l\000i\000t\000\340\000:\000\040\000m\000o\000d\000a\000\040\000e\000\040\000m\000e\000d\000i\000a\000n\000a) +(\376\377\000V\000a\000l\000o\000r\000e\000\040\000a\000t\000t\000e\000s\000o\000\040\000s\000u\000\040\000v\000.\000a\000.\000\040\000i\000n\000t\000e\000g\000r\000a\000b\000i\000l\000i\000\040\000e\000/\000o\000\040\000n\000o\000n\000\040\000n\000e\000g\000a\000t\000i\000v\000e) endobj 152 0 obj -<< /S /GoTo /D (section.2.5) >> +<< /S /GoTo /D (subsection.2.3.2) >> endobj 155 0 obj -(\376\377\000I\000n\000d\000i\000c\000i\000\040\000d\000i\000\040\000d\000i\000s\000p\000e\000r\000s\000i\000o\000n\000e\000:\000\040\000c\000o\000v\000a\000r\000i\000a\000n\000z\000a\000,\000\040\000v\000a\000r\000i\000a\000n\000z\000a\000,\000\040\000d\000e\000v\000.\000\040\000s\000t\000a\000n\000d\000a\000r\000d\000\040\000e\000\040\000c\000o\000e\000f\000f\000.\000\040\000d\000i\000\040\000c\000o\000r\000r\000e\000l\000a\000z\000i\000o\000n\000e) +(\376\377\000P\000r\000o\000p\000r\000i\000e\000t\000\340\000\040\000d\000e\000l\000\040\000v\000a\000l\000o\000r\000e\000\040\000a\000t\000t\000e\000s\000o\000\040\000e\000\040\000m\000o\000l\000t\000i\000p\000l\000i\000c\000a\000t\000i\000v\000i\000t\000\340\000\040\000p\000e\000r\000\040\000v\000.\000a\000.\000\040\000i\000n\000d\000i\000p\000e\000n\000d\000e\000n\000t\000i) endobj 156 0 obj -<< /S /GoTo /D (subsection.2.5.1) >> +<< /S /GoTo /D (subsection.2.3.3) >> endobj 159 0 obj -(\376\377\000D\000e\000f\000i\000n\000i\000z\000i\000o\000n\000i\000\040\000e\000\040\000c\000o\000v\000a\000r\000i\000a\000n\000z\000a\000\040\000c\000o\000m\000e\000\040\000f\000o\000r\000m\000a\000\040\000b\000i\000l\000i\000n\000e\000a\000r\000e\000\040\000s\000i\000m\000m\000e\000t\000r\000i\000c\000a) +(\376\377\000V\000a\000l\000o\000r\000e\000\040\000a\000t\000t\000e\000s\000o\000\040\000c\000o\000n\000d\000i\000z\000i\000o\000n\000a\000l\000e) endobj 160 0 obj -<< /S /GoTo /D (subsection.2.5.2) >> +<< /S /GoTo /D (subsection.2.3.4) >> endobj 163 0 obj -(\376\377\000I\000d\000e\000n\000t\000i\000t\000\340\000\040\000s\000u\000l\000l\000a\000\040\000\050\000c\000o\000\051\000v\000a\000r\000i\000a\000n\000z\000a\000\040\000e\000\040\000d\000i\000s\000u\000g\000u\000a\000g\000l\000i\000a\000n\000z\000a\000\040\000d\000i\000\040\000C\000h\000e\000b\000y\000s\000h\000e\000v) +(\376\377\000M\000o\000m\000e\000n\000t\000i\000\040\000\050\000a\000s\000s\000o\000l\000u\000t\000i\000\051\000\040\000n\000-\000e\000s\000i\000m\000i) endobj 164 0 obj -<< /S /GoTo /D (subsection.2.5.3) >> +<< /S /GoTo /D (subsection.2.3.5) >> endobj 167 0 obj -(\376\377\000C\000o\000e\000f\000f\000.\000\040\000d\000i\000\040\000c\000o\000r\000r\000e\000l\000a\000z\000i\000o\000n\000e\000\040\000e\000\040\000r\000e\000t\000t\000a\000\040\000d\000i\000\040\000r\000e\000g\000r\000e\000s\000s\000i\000o\000n\000e\000\040\000l\000i\000n\000e\000a\000r\000e) +(\376\377\000D\000i\000s\000u\000g\000u\000a\000g\000l\000i\000a\000n\000z\000a\000\040\000d\000i\000\040\000M\000a\000r\000k\000o\000v\000,\000\040\000d\000i\000\040\000H\000\366\000l\000d\000e\000r\000,\000\040\000d\000i\000\040\000C\000a\000u\000c\000h\000y\000-\000S\000c\000h\000w\000a\000r\000z\000\040\000e\000\040\000d\000i\000\040\000J\000e\000n\000s\000e\000n) endobj 168 0 obj -<< /S /GoTo /D (section.2.6) >> +<< /S /GoTo /D (section.2.4) >> endobj 171 0 obj -(\376\377\000L\000e\000g\000g\000e\000\040\000d\000e\000i\000\040\000g\000r\000a\000n\000d\000i\000\040\000n\000u\000m\000e\000r\000i\000\040\000\050\000L\000G\000N\000\051\000,\000\040\000m\000e\000d\000i\000a\000\040\000c\000a\000m\000p\000i\000o\000n\000a\000r\000i\000a\000\040\000e\000\040\000l\000i\000m\000i\000t\000e\000\040\000i\000n\000\040\000s\000e\000n\000s\000o\000\040\000p\000r\000o\000b\000a\000b\000i\000l\000i\000s\000t\000i\000c\000o) +(\376\377\000A\000l\000t\000r\000i\000\040\000i\000n\000d\000i\000c\000i\000\040\000d\000i\000\040\000c\000e\000n\000t\000r\000a\000l\000i\000t\000\340\000:\000\040\000m\000o\000d\000a\000\040\000e\000\040\000m\000e\000d\000i\000a\000n\000a) endobj 172 0 obj -<< /S /GoTo /D (subsection.2.6.1) >> +<< /S /GoTo /D (section.2.5) >> endobj 175 0 obj -(\376\377\000D\000e\000f\000i\000n\000i\000z\000i\000o\000n\000i\000\040\000e\000d\000\040\000e\000n\000u\000n\000c\000i\000a\000t\000o) +(\376\377\000I\000n\000d\000i\000c\000i\000\040\000d\000i\000\040\000d\000i\000s\000p\000e\000r\000s\000i\000o\000n\000e\000:\000\040\000c\000o\000v\000a\000r\000i\000a\000n\000z\000a\000,\000\040\000v\000a\000r\000i\000a\000n\000z\000a\000,\000\040\000d\000e\000v\000.\000\040\000s\000t\000a\000n\000d\000a\000r\000d\000\040\000e\000\040\000c\000o\000e\000f\000f\000.\000\040\000d\000i\000\040\000c\000o\000r\000r\000e\000l\000a\000z\000i\000o\000n\000e) endobj 176 0 obj -<< /S /GoTo /D (subsection.2.6.2) >> +<< /S /GoTo /D (subsection.2.5.1) >> endobj 179 0 obj -(\376\377\000T\000r\000a\000s\000f\000o\000r\000m\000a\000t\000a\000\040\000d\000i\000\040\000C\000r\000a\000m\000e\000r\000\040\000p\000e\000r\000\040\000l\000'\000o\000t\000t\000i\000m\000i\000z\000z\000a\000z\000i\000o\000n\000e\000\040\000d\000e\000l\000l\000a\000\040\000s\000t\000i\000m\000a) 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/GoTo /D (subsection.2.7.1) >> +<< /S /GoTo /D (subsection.2.5.3) >> endobj 187 0 obj -(\376\377\000I\000n\000t\000u\000i\000z\000i\000o\000n\000e\000\040\000d\000e\000l\000\040\000T\000C\000L\000:\000\040\000z\000o\000o\000m\000-\000i\000n\000\040\000e\000\040\000s\000c\000a\000l\000i\000n\000g) +(\376\377\000C\000o\000e\000f\000f\000.\000\040\000d\000i\000\040\000c\000o\000r\000r\000e\000l\000a\000z\000i\000o\000n\000e\000\040\000e\000\040\000r\000e\000t\000t\000a\000\040\000d\000i\000\040\000r\000e\000g\000r\000e\000s\000s\000i\000o\000n\000e\000\040\000l\000i\000n\000e\000a\000r\000e) endobj 188 0 obj -<< /S /GoTo /D (subsection.2.7.2) >> +<< /S /GoTo /D (section.2.6) >> endobj 191 0 obj 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probabilità e statistica/main.tex b/Secondo anno/Elementi di probabilità e statistica/main.tex index ec10e6a..62f1f20 100644 --- a/Secondo anno/Elementi di probabilità e statistica/main.tex +++ b/Secondo anno/Elementi di probabilità e statistica/main.tex @@ -11,7 +11,7 @@ \title{\Huge{Schede riassuntive di \\ \textit{Elementi di Probabilità e Statistica}}} -\date{A.A. 2023-2024 \\[0.6in] Ultimo aggiornamento: \today} +\date{A.A. 2023-2024 \\[0.6in] Ultimo aggiornamento: \today \\[1in] \href{https://eps.hearot.it}{\texttt{https://eps.hearot.it}}} \author{A cura di Gabriel Antonio Videtta\footnote{Basato su un layout di \underline{Luca Lombardo} e di \underline{Francesco Sorce}.} \\ \href{mailto:g.videtta1@studenti.unipi.it}{\texttt{g.videtta1@studenti.unipi.it}}} \begin{document} @@ -27,6 +27,7 @@ \input{sections/0-identità-somme.tex} \input{sections/1-spazi-in-generale.tex} \input{sections/2-probabilità-discreta.tex} +\input{sections/3-probabilità-reale.tex} \input{sections/tabella-modelli-discreti.tex} \input{sections/tabella-phi.tex} diff --git a/Secondo anno/Elementi di probabilità e statistica/preamble.tex b/Secondo anno/Elementi di probabilità e statistica/preamble.tex index 4c0fc4e..9fe1e15 100644 --- a/Secondo anno/Elementi di probabilità e statistica/preamble.tex +++ b/Secondo anno/Elementi di probabilità e statistica/preamble.tex @@ -120,7 +120,7 @@ \newcommand{\qc}{q.c.\ \!} \newcommand{\va}{v.a.\ \!} - +\newcommand{\BB}{\mathcal{B}} \newcommand{\EE}{\mathbb{E}} \DeclareMathOperator{\Var}{Var} \DeclareMathOperator{\Cov}{Cov} 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 8d7939f..a761346 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 @@ -6,6 +6,7 @@ \begin{multicols*}{2} \section*{Algebra lineare} +\addcontentsline{toc}{section}{Algebra lineare} \begin{itemize} \item \textbf{Disuguaglianza di Cauchy-Schwarz} -- Se $\varphi(\cdot, \cdot)$ @@ -22,6 +23,7 @@ \end{itemize} \section*{Analisi matematica} +\addcontentsline{toc}{section}{Analisi matematica} \begin{itemize} \item \textbf{Limite delle successioni monotone} -- Se una successione @@ -64,6 +66,7 @@ \end{itemize} \section*{Combinatoria} +\addcontentsline{toc}{section}{Combinatoria} \begin{itemize} \item \textbf{Principio di \textit{double counting}} -- Principio di dimostrazione per il quale @@ -122,6 +125,7 @@ \end{itemize} \section*{Teoria degli insiemi} +\addcontentsline{toc}{section}{Teoria degli insiemi} \begin{itemize} \item \textbf{Leggi di De Morgan} -- Se $A$ e $B$ sono insiemi, allora @@ -135,4 +139,15 @@ per $A \subseteq C'$. \end{itemize} +\section*{Teoria della misura} +\addcontentsline{toc}{section}{Teoria della misura} + +Per quanto riguarda teoria della misura si è preferito +riportare i risultati principali dati a lezione anche sotto la +\textit{Parte 3}. + +% \begin{itemize} +% \item +% \end{itemize} + \end{multicols*} \ No newline at end of file diff --git a/Secondo anno/Elementi di probabilità e statistica/sections/1-spazi-in-generale.tex b/Secondo anno/Elementi di probabilità e statistica/sections/1-spazi-in-generale.tex index d4231b4..bb6d810 100644 --- a/Secondo anno/Elementi di probabilità e statistica/sections/1-spazi-in-generale.tex +++ b/Secondo anno/Elementi di probabilità e statistica/sections/1-spazi-in-generale.tex @@ -119,7 +119,7 @@ \item $P(\bigcupdot_{i \in \NN} A_i) = \sum_{i \in \NN} P(A_i)$ ($\sigma$-additività). \end{enumerate} - In particolare $P$ è una misura. + In particolare $P$ è una misura per cui $P(\Omega) = 1$. \end{definition} \begin{definition}[Spazio di probabilità] 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 new file mode 100644 index 0000000..5c89201 --- /dev/null +++ b/Secondo anno/Elementi di probabilità e statistica/sections/3-probabilità-reale.tex @@ -0,0 +1,78 @@ +%-------------------------------------------------------------------- +\chapter{Probabilità sulla retta reale} +\setlength{\parindent}{2pt} + +\begin{multicols*}{2} + +Discutiamo in questa sezione la teoria della probabilità sulla +retta reale, uscendo dunque dal caso discreto. + +Per restringere la $\sigma$-algebra su cui lavoreremo (ossia l'insieme degli +eventi interessanti), siamo costretti a limitarci a una $\sigma$-algebra molto più +piccola di $\PP(\RR)$, la $\sigma$-algebra dei boreliani, che ci permette di escludere +``casi meno interessanti''. + +\section{Cenni di teoria della misura} + +\subsection{La \texorpdfstring{$\sigma$}{σ}-algebra di Borel} + +\begin{definition}[$\sigma$-algebra dei boreliani] + Dato uno spazio metrico separabile\footnote{ + Si può generalizzare in modo naturale tale definizione a un qualsiasi spazio topologico. + Dal momento che considereremo solo spazi metrici separabili (in particolare $X \subseteq \RR^d$), concentreremo + le proprietà e le definizioni su questa classe di spazi topologici.} $X \neq \emptyset$ + si definisce la \textbf{$\sigma$-algebra $\BB(X)$ dei boreliani di $X$} (o + $\sigma$-algebra di Borel) + come la $\sigma$-algebra generata dai suoi aperti, ovverosia: + \[ + \BB(X) \defeq \sigma \{ A \subseteq X \mid A \text{ aperto}\, \}. + \] +\end{definition} + +\begin{proposition}[Proprietà di $\BB(X)$] + Sia $X \neq \emptyset$ uno spazio metrico separabile. Allora valgono + le seguenti affermazioni: + + \begin{enumerate}[(i.)] + \item $\BB(X)$ contiene tutti gli aperti e i chiusi di $X$ (infatti + metrico e separabile implica II-numerabile), + \item $\BB(X) = \sigma \{ A \subseteq X \mid A \text{ chiuso}\, \}$, ossia + $\BB(X)$ è generata anche dai chiusi di $X$ (infatti $\BB(X)$ è chiuso per + complementare), + \item se $Y \subseteq X$, $Y \neq \emptyset$ ha metrica indotta da $X$, allora + $\BB(Y) = \sigma \{ Y \cap B \mid B \in \BB(X) \} \subseteq \BB(X)$ (segue dal fatto che + gli aperti di $Y$ sono tutti e solo gli aperti di $X$ intersecati a $Y$). + \end{enumerate} +\end{proposition} + +\begin{proposition}[Proprietà di $\BB(\RR^d)$] + Valgono le seguenti affermazioni: + \begin{enumerate}[(i.)] + \item $\BB(\RR)$ contiene tutti gli intervalli e tutte le semirette (infatti si ammettono anche intersezioni infinite di aperti), + \item $\BB(\RR)$ è generato dagli intervalli semiaperti, ovverosia $\BB(\RR) = \sigma \{ (a, b] \mid a, b \in \RR, a < b \}$, + \item $\BB(\RR)$ è generato dalle semirette, ovverosia $\BB(\RR) = \sigma \{ (-\infty, a) \mid a \in \RR \}$, + \item $\BB(\RR^d) = \sigma \{ (-\infty, a_1) \times \ldots \times (-\infty, a_n) \mid a_1, \ldots, a_n \in \RR \}$ (segue da (iii.)), + \item $\BB(\RR^d) \neq \PP(\RR^d)$ (segue dal controesempio di Vitali, oltre che da considerazioni sulle cardinalità). + \end{enumerate} +\end{proposition} + +\subsection{Definizione di misura e misura di Lebesgue} + +\begin{definition}[Misura] + Dato $(\Omega, \FF)$ spazio misurabile, una misura $\mu$ su $(\Omega, \FF)$ è una + funzione $\mu : \FF \to [0, \infty]$ con $\mu(\emptyset) = 0$ e per cui valga + la $\sigma$-additività, ovverosia: + \[ + \mu\left(\bigcupdot_{i \in \NN} A_i\right) = \sum_{i \in \NN} \mu(A_i), \quad A_i \in \FF. + \] +\end{definition} + +\begin{definition}[Insiemi $\mu$-trascurabili e proprietà $\mu$-quasi certe] + Un insieme $A \in \FF$ si dice \textbf{$\mu$-trascurabile} se + $\mu(A) = 0$. Una proprietà $M$ si dice che accade + $\mu$-quasi certamente se esiste $A \in \FF$ $\mu$-trascurabile per cui + $M$ accade per $A^c$. +\end{definition} + + +\end{multicols*} \ No newline at end of file