From 4533f38ea40c9fa7fafdcc4a8c748514ab965087 Mon Sep 17 00:00:00 2001 From: Hearot Date: Mon, 22 Apr 2024 22:36:24 +0200 Subject: [PATCH] feat(eps): tabella distribuzioni assolutamente continue --- .../main.pdf | 5283 +++++++++-------- .../main.tex | 1 + .../preamble.tex | 6 +- .../sections/0-notazioni.tex | 1 + .../sections/3-probabilità-reale.tex | 8 +- .../sections/tabella-modelli-reali.tex | 34 + .../sections/tabella-phi.tex | 2 +- 7 files changed, 2768 insertions(+), 2567 deletions(-) create mode 100644 Secondo anno/Elementi di probabilità e statistica/sections/tabella-modelli-reali.tex diff --git a/Secondo anno/Elementi di probabilità e statistica/main.pdf b/Secondo anno/Elementi di probabilità e statistica/main.pdf index 3e31a13..81410e5 100644 --- a/Secondo anno/Elementi di probabilità e statistica/main.pdf +++ b/Secondo anno/Elementi di probabilità e statistica/main.pdf @@ -454,12 +454,18 @@ endobj << /S /GoTo /D (chapter*.18) >> endobj 307 0 obj -(\376\377\000T\000a\000b\000e\000l\000l\000a\000\040\000e\000\040\000p\000r\000o\000p\000r\000i\000e\000t\000\340\000\040\000d\000e\000l\000l\000a\000\040\000f\000.\000d\000.\000r\000.\000\040\003\246\000\050\000x\000\051\000\040\000d\000e\000l\000l\000a\000\040\000n\000o\000r\000m\000a\000l\000e\000\040\000s\000t\000a\000n\000d\000a\000r\000d\000\040\000N\000\050\0000\000,\000\040\0001\000\051) +(\376\377\000T\000a\000b\000e\000l\000l\000a\000\040\000e\000\040\000p\000r\000o\000p\000r\000i\000e\000t\000\340\000\040\000d\000e\000l\000l\000e\000\040\000d\000i\000s\000t\000r\000i\000b\000u\000z\000i\000o\000n\000i\000\040\000a\000s\000s\000o\000l\000u\000t\000a\000m\000e\000n\000t\000e\000\040\000c\000o\000n\000t\000i\000n\000u\000e) endobj 308 0 obj -<< /S /GoTo /D [309 0 R /Fit] >> +<< /S /GoTo /D (chapter*.19) >> endobj -313 0 obj +311 0 obj +(\376\377\000T\000a\000b\000e\000l\000l\000a\000\040\000e\000\040\000p\000r\000o\000p\000r\000i\000e\000t\000\340\000\040\000d\000e\000l\000l\000a\000\040\000f\000.\000d\000.\000r\000.\000\040\003\246\000\050\000x\000\051\000\040\000d\000e\000l\000l\000a\000\040\000n\000o\000r\000m\000a\000l\000e\000\040\000s\000t\000a\000n\000d\000a\000r\000d\000\040\000N\000\050\0000\000,\000\040\0001\000\051) +endobj +312 0 obj +<< /S /GoTo /D [313 0 R /Fit] >> +endobj +317 0 obj << /Length 777 /Filter /FlateDecode @@ -469,17 +475,17 @@ xڅTMo p󛫫0X4u6f^8}֬sP,o_E_(xnZ06uL3=ZFU&ONE>@i'SBnw%Jv- ݯ|ǫ /c?i7;vi?'a=SZ.! 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(theorem.1.8) 473 0 R (theorem.1.9) 477 0 R] -/Limits [(theorem.1.44) (theorem.1.9)] +/Names [(section.2.5) 197 0 R (section.2.6) 213 0 R (section.2.7) 225 0 R (section.2.8) 237 0 R (section.3.1) 253 0 R (section.3.2) 269 0 R] +/Limits [(section.2.5) (section.3.2)] >> endobj 936 0 obj << -/Names [(theorem.2.1) 544 0 R (theorem.2.10) 555 0 R (theorem.2.100) 704 0 R (theorem.2.101) 705 0 R (theorem.2.102) 713 0 R (theorem.2.103) 716 0 R] -/Limits [(theorem.2.1) (theorem.2.103)] +/Names [(section.3.3) 281 0 R (section.3.4) 285 0 R (section.3.5) 289 0 R (subsection.1.1.1) 69 0 R (subsection.1.1.2) 73 0 R (subsection.1.1.3) 77 0 R] +/Limits [(section.3.3) (subsection.1.1.3)] >> endobj 937 0 obj << -/Names [(theorem.2.104) 717 0 R (theorem.2.105) 718 0 R (theorem.2.106) 719 0 R (theorem.2.107) 720 0 R (theorem.2.108) 721 0 R (theorem.2.11) 556 0 R] -/Limits [(theorem.2.104) (theorem.2.11)] +/Names [(subsection.1.1.4) 81 0 R (subsection.1.3.1) 93 0 R (subsection.1.3.2) 97 0 R (subsection.1.3.3) 101 0 R (subsection.1.4.1) 109 0 R (subsection.1.4.2) 113 0 R] +/Limits [(subsection.1.1.4) (subsection.1.4.2)] >> endobj 938 0 obj << -/Names [(theorem.2.12) 557 0 R (theorem.2.13) 564 0 R (theorem.2.14) 565 0 R (theorem.2.15) 566 0 R (theorem.2.16) 567 0 R (theorem.2.17) 568 0 R] -/Limits [(theorem.2.12) (theorem.2.17)] +/Names [(subsection.1.4.3) 117 0 R (subsection.2.1.1) 133 0 R (subsection.2.1.2) 137 0 R (subsection.2.1.3) 141 0 R (subsection.2.2.1) 149 0 R (subsection.2.2.2) 153 0 R] +/Limits [(subsection.1.4.3) (subsection.2.2.2)] >> endobj 939 0 obj << -/Names [(theorem.2.18) 569 0 R (theorem.2.19) 570 0 R (theorem.2.2) 545 0 R (theorem.2.20) 571 0 R (theorem.2.21) 572 0 R (theorem.2.22) 573 0 R] -/Limits [(theorem.2.18) (theorem.2.22)] +/Names [(subsection.2.2.3) 157 0 R (subsection.2.2.4) 161 0 R (subsection.2.2.5) 165 0 R (subsection.2.3.1) 173 0 R (subsection.2.3.2) 177 0 R (subsection.2.3.3) 181 0 R] +/Limits [(subsection.2.2.3) (subsection.2.3.3)] >> endobj 940 0 obj << -/Names [(theorem.2.23) 574 0 R (theorem.2.24) 575 0 R (theorem.2.25) 576 0 R (theorem.2.26) 577 0 R (theorem.2.27) 584 0 R (theorem.2.28) 585 0 R] -/Limits [(theorem.2.23) (theorem.2.28)] +/Names [(subsection.2.3.4) 185 0 R (subsection.2.3.5) 189 0 R (subsection.2.5.1) 201 0 R (subsection.2.5.2) 205 0 R (subsection.2.5.3) 209 0 R (subsection.2.6.1) 217 0 R] +/Limits [(subsection.2.3.4) (subsection.2.6.1)] >> endobj 941 0 obj << -/Names [(theorem.2.29) 586 0 R (theorem.2.3) 548 0 R (theorem.2.30) 587 0 R (theorem.2.31) 588 0 R (theorem.2.32) 589 0 R (theorem.2.33) 590 0 R] -/Limits [(theorem.2.29) (theorem.2.33)] +/Names [(subsection.2.6.2) 221 0 R (subsection.2.7.1) 229 0 R (subsection.2.7.2) 233 0 R (subsection.2.8.1) 241 0 R (subsection.2.8.2) 245 0 R (subsection.3.1.1) 257 0 R] +/Limits [(subsection.2.6.2) (subsection.3.1.1)] >> endobj 942 0 obj << -/Names [(theorem.2.34) 591 0 R (theorem.2.35) 592 0 R (theorem.2.36) 593 0 R (theorem.2.37) 594 0 R (theorem.2.38) 596 0 R (theorem.2.39) 597 0 R] -/Limits [(theorem.2.34) (theorem.2.39)] +/Names [(subsection.3.1.2) 261 0 R (subsection.3.1.3) 265 0 R (subsection.3.2.1) 273 0 R (subsection.3.2.2) 277 0 R (subsection.3.5.1) 293 0 R (subsection.3.5.2) 297 0 R] +/Limits [(subsection.3.1.2) (subsection.3.5.2)] >> endobj 943 0 obj << -/Names [(theorem.2.4) 549 0 R (theorem.2.40) 598 0 R (theorem.2.41) 599 0 R (theorem.2.42) 606 0 R (theorem.2.43) 607 0 R (theorem.2.44) 608 0 R] -/Limits [(theorem.2.4) (theorem.2.44)] +/Names [(table.3.1) 733 0 R (theorem.1.1) 474 0 R (theorem.1.10) 493 0 R (theorem.1.11) 497 0 R (theorem.1.12) 498 0 R (theorem.1.13) 507 0 R] +/Limits [(table.3.1) (theorem.1.13)] >> endobj 944 0 obj << -/Names [(theorem.2.45) 609 0 R (theorem.2.46) 612 0 R (theorem.2.47) 613 0 R (theorem.2.48) 614 0 R (theorem.2.49) 615 0 R (theorem.2.5) 550 0 R] -/Limits [(theorem.2.45) (theorem.2.5)] +/Names [(theorem.1.14) 508 0 R (theorem.1.15) 511 0 R (theorem.1.16) 512 0 R (theorem.1.17) 513 0 R (theorem.1.18) 514 0 R (theorem.1.19) 515 0 R] +/Limits [(theorem.1.14) (theorem.1.19)] >> endobj 945 0 obj << -/Names [(theorem.2.50) 616 0 R (theorem.2.51) 617 0 R (theorem.2.52) 618 0 R (theorem.2.53) 621 0 R (theorem.2.54) 629 0 R (theorem.2.55) 636 0 R] -/Limits [(theorem.2.50) (theorem.2.55)] +/Names [(theorem.1.2) 475 0 R (theorem.1.20) 516 0 R (theorem.1.21) 517 0 R (theorem.1.22) 518 0 R (theorem.1.23) 519 0 R (theorem.1.24) 520 0 R] +/Limits [(theorem.1.2) (theorem.1.24)] >> endobj 946 0 obj << -/Names [(theorem.2.56) 637 0 R (theorem.2.57) 638 0 R (theorem.2.58) 639 0 R (theorem.2.59) 640 0 R (theorem.2.6) 551 0 R (theorem.2.60) 641 0 R] -/Limits [(theorem.2.56) (theorem.2.60)] +/Names [(theorem.1.25) 521 0 R (theorem.1.26) 527 0 R (theorem.1.27) 528 0 R (theorem.1.28) 529 0 R (theorem.1.29) 530 0 R (theorem.1.3) 476 0 R] +/Limits [(theorem.1.25) (theorem.1.3)] >> endobj 947 0 obj << -/Names [(theorem.2.61) 642 0 R (theorem.2.62) 643 0 R (theorem.2.63) 644 0 R (theorem.2.64) 645 0 R (theorem.2.65) 646 0 R (theorem.2.66) 647 0 R] -/Limits [(theorem.2.61) (theorem.2.66)] +/Names [(theorem.1.30) 531 0 R (theorem.1.31) 532 0 R (theorem.1.32) 533 0 R (theorem.1.33) 534 0 R (theorem.1.34) 536 0 R (theorem.1.35) 537 0 R] +/Limits [(theorem.1.30) (theorem.1.35)] >> endobj 948 0 obj << -/Names [(theorem.2.67) 655 0 R (theorem.2.68) 656 0 R (theorem.2.69) 657 0 R (theorem.2.7) 552 0 R (theorem.2.70) 658 0 R (theorem.2.71) 659 0 R] -/Limits [(theorem.2.67) (theorem.2.71)] +/Names [(theorem.1.36) 538 0 R (theorem.1.37) 539 0 R (theorem.1.38) 540 0 R (theorem.1.39) 541 0 R (theorem.1.4) 480 0 R (theorem.1.40) 542 0 R] +/Limits [(theorem.1.36) (theorem.1.40)] >> endobj 949 0 obj << -/Names [(theorem.2.72) 660 0 R (theorem.2.73) 661 0 R (theorem.2.74) 662 0 R (theorem.2.75) 663 0 R (theorem.2.76) 664 0 R (theorem.2.77) 665 0 R] -/Limits [(theorem.2.72) (theorem.2.77)] +/Names [(theorem.1.41) 547 0 R (theorem.1.42) 548 0 R (theorem.1.43) 549 0 R (theorem.1.44) 550 0 R (theorem.1.5) 481 0 R (theorem.1.6) 482 0 R] +/Limits [(theorem.1.41) (theorem.1.6)] >> endobj 950 0 obj << -/Names [(theorem.2.78) 666 0 R (theorem.2.79) 667 0 R (theorem.2.8) 553 0 R (theorem.2.80) 668 0 R (theorem.2.81) 669 0 R (theorem.2.82) 678 0 R] -/Limits [(theorem.2.78) (theorem.2.82)] +/Names [(theorem.1.7) 483 0 R (theorem.1.8) 484 0 R (theorem.1.9) 488 0 R (theorem.2.1) 556 0 R (theorem.2.10) 567 0 R (theorem.2.100) 715 0 R] +/Limits [(theorem.1.7) (theorem.2.100)] >> endobj 951 0 obj << -/Names [(theorem.2.83) 679 0 R (theorem.2.84) 680 0 R (theorem.2.85) 681 0 R (theorem.2.86) 682 0 R (theorem.2.87) 683 0 R (theorem.2.88) 684 0 R] -/Limits [(theorem.2.83) (theorem.2.88)] +/Names [(theorem.2.101) 716 0 R (theorem.2.102) 724 0 R (theorem.2.103) 726 0 R (theorem.2.104) 727 0 R (theorem.2.105) 728 0 R (theorem.2.106) 729 0 R] +/Limits [(theorem.2.101) (theorem.2.106)] >> endobj 952 0 obj << -/Names [(theorem.2.89) 685 0 R (theorem.2.9) 554 0 R (theorem.2.90) 686 0 R (theorem.2.91) 687 0 R (theorem.2.92) 690 0 R (theorem.2.93) 691 0 R] -/Limits [(theorem.2.89) (theorem.2.93)] +/Names [(theorem.2.107) 730 0 R (theorem.2.108) 731 0 R (theorem.2.11) 568 0 R (theorem.2.12) 569 0 R (theorem.2.13) 575 0 R (theorem.2.14) 576 0 R] +/Limits [(theorem.2.107) (theorem.2.14)] >> endobj 953 0 obj << -/Names [(theorem.2.94) 692 0 R (theorem.2.95) 699 0 R (theorem.2.96) 700 0 R (theorem.2.97) 701 0 R (theorem.2.98) 702 0 R (theorem.2.99) 703 0 R] -/Limits [(theorem.2.94) (theorem.2.99)] +/Names [(theorem.2.15) 577 0 R (theorem.2.16) 578 0 R (theorem.2.17) 579 0 R (theorem.2.18) 580 0 R (theorem.2.19) 581 0 R (theorem.2.2) 557 0 R] +/Limits [(theorem.2.15) (theorem.2.2)] >> endobj 954 0 obj << -/Names [(theorem.3.1) 733 0 R (theorem.3.10) 757 0 R (theorem.3.11) 764 0 R (theorem.3.12) 765 0 R (theorem.3.13) 768 0 R (theorem.3.14) 769 0 R] -/Limits [(theorem.3.1) (theorem.3.14)] +/Names [(theorem.2.20) 582 0 R (theorem.2.21) 583 0 R (theorem.2.22) 584 0 R (theorem.2.23) 585 0 R (theorem.2.24) 586 0 R (theorem.2.25) 587 0 R] +/Limits [(theorem.2.20) (theorem.2.25)] >> endobj 955 0 obj << -/Names [(theorem.3.15) 770 0 R (theorem.3.16) 771 0 R (theorem.3.17) 772 0 R (theorem.3.18) 773 0 R (theorem.3.19) 774 0 R (theorem.3.2) 734 0 R] -/Limits [(theorem.3.15) (theorem.3.2)] +/Names [(theorem.2.26) 588 0 R (theorem.2.27) 595 0 R (theorem.2.28) 596 0 R (theorem.2.29) 597 0 R (theorem.2.3) 560 0 R (theorem.2.30) 598 0 R] +/Limits [(theorem.2.26) (theorem.2.30)] >> endobj 956 0 obj << -/Names [(theorem.3.20) 775 0 R (theorem.3.21) 787 0 R (theorem.3.22) 792 0 R (theorem.3.23) 793 0 R (theorem.3.24) 794 0 R (theorem.3.25) 795 0 R] -/Limits [(theorem.3.20) (theorem.3.25)] +/Names [(theorem.2.31) 599 0 R (theorem.2.32) 600 0 R (theorem.2.33) 601 0 R (theorem.2.34) 602 0 R (theorem.2.35) 603 0 R (theorem.2.36) 604 0 R] +/Limits [(theorem.2.31) (theorem.2.36)] >> endobj 957 0 obj << -/Names [(theorem.3.26) 796 0 R (theorem.3.27) 797 0 R (theorem.3.28) 798 0 R (theorem.3.29) 804 0 R (theorem.3.3) 738 0 R (theorem.3.30) 805 0 R] -/Limits [(theorem.3.26) (theorem.3.30)] +/Names [(theorem.2.37) 605 0 R (theorem.2.38) 607 0 R (theorem.2.39) 608 0 R (theorem.2.4) 561 0 R (theorem.2.40) 609 0 R (theorem.2.41) 610 0 R] +/Limits [(theorem.2.37) (theorem.2.41)] >> endobj 958 0 obj << -/Names [(theorem.3.31) 806 0 R (theorem.3.32) 807 0 R (theorem.3.33) 808 0 R (theorem.3.34) 809 0 R (theorem.3.35) 812 0 R (theorem.3.36) 815 0 R] -/Limits [(theorem.3.31) (theorem.3.36)] +/Names [(theorem.2.42) 617 0 R (theorem.2.43) 618 0 R (theorem.2.44) 619 0 R (theorem.2.45) 620 0 R (theorem.2.46) 623 0 R (theorem.2.47) 624 0 R] +/Limits [(theorem.2.42) (theorem.2.47)] >> endobj 959 0 obj << -/Names [(theorem.3.4) 744 0 R (theorem.3.5) 745 0 R (theorem.3.6) 746 0 R (theorem.3.7) 747 0 R (theorem.3.8) 753 0 R (theorem.3.9) 756 0 R] -/Limits [(theorem.3.4) (theorem.3.9)] +/Names [(theorem.2.48) 625 0 R (theorem.2.49) 626 0 R (theorem.2.5) 562 0 R (theorem.2.50) 627 0 R (theorem.2.51) 628 0 R (theorem.2.52) 629 0 R] +/Limits [(theorem.2.48) (theorem.2.52)] >> endobj 960 0 obj << -/Kids [899 0 R 900 0 R 901 0 R 902 0 R 903 0 R 904 0 R] -/Limits [(Doc-Start) (Item.33)] +/Names [(theorem.2.53) 632 0 R (theorem.2.54) 640 0 R (theorem.2.55) 647 0 R (theorem.2.56) 648 0 R (theorem.2.57) 649 0 R (theorem.2.58) 650 0 R] +/Limits [(theorem.2.53) (theorem.2.58)] >> endobj 961 0 obj << -/Kids [905 0 R 906 0 R 907 0 R 908 0 R 909 0 R 910 0 R] -/Limits [(Item.34) (Item.9)] +/Names [(theorem.2.59) 651 0 R (theorem.2.6) 563 0 R (theorem.2.60) 652 0 R (theorem.2.61) 653 0 R (theorem.2.62) 654 0 R (theorem.2.63) 655 0 R] +/Limits [(theorem.2.59) (theorem.2.63)] >> endobj 962 0 obj << -/Kids [911 0 R 912 0 R 913 0 R 914 0 R 915 0 R 916 0 R] -/Limits [(chapter*.1) (page.9)] +/Names [(theorem.2.64) 656 0 R (theorem.2.65) 657 0 R (theorem.2.66) 658 0 R (theorem.2.67) 666 0 R (theorem.2.68) 667 0 R (theorem.2.69) 668 0 R] +/Limits [(theorem.2.64) (theorem.2.69)] >> endobj 963 0 obj << -/Kids [917 0 R 918 0 R 919 0 R 920 0 R 921 0 R 922 0 R] -/Limits [(section*.10) (subsection.1.3.2)] +/Names [(theorem.2.7) 564 0 R (theorem.2.70) 669 0 R (theorem.2.71) 670 0 R (theorem.2.72) 671 0 R (theorem.2.73) 672 0 R (theorem.2.74) 673 0 R] +/Limits [(theorem.2.7) (theorem.2.74)] >> endobj 964 0 obj << -/Kids [923 0 R 924 0 R 925 0 R 926 0 R 927 0 R 928 0 R] -/Limits [(subsection.1.3.3) (theorem.1.10)] +/Names [(theorem.2.75) 674 0 R (theorem.2.76) 675 0 R (theorem.2.77) 676 0 R (theorem.2.78) 677 0 R (theorem.2.79) 678 0 R (theorem.2.8) 565 0 R] +/Limits [(theorem.2.75) (theorem.2.8)] >> endobj 965 0 obj << -/Kids [929 0 R 930 0 R 931 0 R 932 0 R 933 0 R 934 0 R] -/Limits [(theorem.1.11) (theorem.1.43)] +/Names [(theorem.2.80) 679 0 R (theorem.2.81) 680 0 R (theorem.2.82) 690 0 R (theorem.2.83) 691 0 R (theorem.2.84) 692 0 R (theorem.2.85) 693 0 R] +/Limits [(theorem.2.80) (theorem.2.85)] >> endobj 966 0 obj << -/Kids [935 0 R 936 0 R 937 0 R 938 0 R 939 0 R 940 0 R] -/Limits [(theorem.1.44) (theorem.2.28)] +/Names [(theorem.2.86) 694 0 R (theorem.2.87) 695 0 R (theorem.2.88) 696 0 R (theorem.2.89) 697 0 R (theorem.2.9) 566 0 R (theorem.2.90) 698 0 R] +/Limits [(theorem.2.86) (theorem.2.90)] >> endobj 967 0 obj << -/Kids [941 0 R 942 0 R 943 0 R 944 0 R 945 0 R 946 0 R] -/Limits [(theorem.2.29) (theorem.2.60)] +/Names [(theorem.2.91) 699 0 R (theorem.2.92) 702 0 R (theorem.2.93) 703 0 R (theorem.2.94) 704 0 R (theorem.2.95) 710 0 R (theorem.2.96) 711 0 R] +/Limits [(theorem.2.91) (theorem.2.96)] >> endobj 968 0 obj << -/Kids [947 0 R 948 0 R 949 0 R 950 0 R 951 0 R 952 0 R] -/Limits [(theorem.2.61) (theorem.2.93)] +/Names [(theorem.2.97) 712 0 R (theorem.2.98) 713 0 R (theorem.2.99) 714 0 R (theorem.3.1) 743 0 R (theorem.3.10) 767 0 R (theorem.3.11) 774 0 R] +/Limits [(theorem.2.97) (theorem.3.11)] >> endobj 969 0 obj << -/Kids [953 0 R 954 0 R 955 0 R 956 0 R 957 0 R 958 0 R] -/Limits [(theorem.2.94) (theorem.3.36)] +/Names [(theorem.3.12) 775 0 R (theorem.3.13) 778 0 R (theorem.3.14) 779 0 R (theorem.3.15) 780 0 R (theorem.3.16) 781 0 R (theorem.3.17) 782 0 R] +/Limits [(theorem.3.12) (theorem.3.17)] >> endobj 970 0 obj << -/Kids [959 0 R] -/Limits [(theorem.3.4) (theorem.3.9)] +/Names [(theorem.3.18) 783 0 R (theorem.3.19) 784 0 R (theorem.3.2) 744 0 R (theorem.3.20) 785 0 R (theorem.3.21) 798 0 R (theorem.3.22) 803 0 R] +/Limits [(theorem.3.18) (theorem.3.22)] >> endobj 971 0 obj << -/Kids [960 0 R 961 0 R 962 0 R 963 0 R 964 0 R 965 0 R] -/Limits [(Doc-Start) (theorem.1.43)] +/Names [(theorem.3.23) 804 0 R (theorem.3.24) 805 0 R (theorem.3.25) 806 0 R (theorem.3.26) 807 0 R (theorem.3.27) 808 0 R (theorem.3.28) 809 0 R] +/Limits [(theorem.3.23) (theorem.3.28)] >> endobj 972 0 obj << -/Kids [966 0 R 967 0 R 968 0 R 969 0 R 970 0 R] -/Limits [(theorem.1.44) (theorem.3.9)] +/Names [(theorem.3.29) 814 0 R (theorem.3.3) 748 0 R (theorem.3.30) 815 0 R (theorem.3.31) 816 0 R (theorem.3.32) 817 0 R 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00000 n trailer -<< /Size 977 -/Root 975 0 R -/Info 976 0 R -/ID [ ] >> +<< /Size 992 +/Root 990 0 R +/Info 991 0 R +/ID [ ] >> startxref -385208 +391062 %%EOF diff --git a/Secondo anno/Elementi di probabilità e statistica/main.tex b/Secondo anno/Elementi di probabilità e statistica/main.tex index ca34a33..f83bce5 100644 --- a/Secondo anno/Elementi di probabilità e statistica/main.tex +++ b/Secondo anno/Elementi di probabilità e statistica/main.tex @@ -29,6 +29,7 @@ \input{sections/2-probabilità-discreta.tex} \input{sections/3-probabilità-reale.tex} \input{sections/tabella-modelli-discreti.tex} +\input{sections/tabella-modelli-reali.tex} \input{sections/tabella-phi.tex} \end{document} diff --git a/Secondo anno/Elementi di probabilità e statistica/preamble.tex b/Secondo anno/Elementi di probabilità e statistica/preamble.tex index f45e0b5..3990d45 100644 --- a/Secondo anno/Elementi di probabilità e statistica/preamble.tex +++ b/Secondo anno/Elementi di probabilità e statistica/preamble.tex @@ -88,12 +88,16 @@ \DeclareMathOperator{\id}{id} \DeclareMathOperator{\sgn}{sgn} +\DeclareMathOperator{\Exp}{Exp} + \DeclareMathOperator{\CI}{CI} \newcommand{\eps}{\varepsilon} \newcommand*\dif{\mathop{}\!\textnormal{\slshape d}} \newcommand{\dx}{\dif{x}} -\newcommand{\dy}{\dif{dy}} +\newcommand{\dy}{\dif{y}} +\newcommand{\dz}{\dif{z}} +\newcommand{\dt}{\dif{t}} %\setcounter{secnumdepth}{1} 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 19a75d2..00ea26c 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 @@ -68,6 +68,7 @@ \item $\log \equiv \ln = \log_e$ -- Logaritmo naturale, ossia logaritmo in base $e$. \item $C^n$ -- Classe di funzioni derivabili $n$ volte con $n$-esima derivata continua. Per $n = 0$, classe di funzioni continue. \item $C^\infty$ -- Classe di funzioni derivabili un numero illimitato di volte. + \item $\Gamma(x) = \int_0^\infty t^{x-1} e^{-t} \dt$ -- Funzione gamma. È tale per cui $\Gamma(n+1) = n!$ per ogni $n \in \NN$. \end{itemize} \section*{Combinatoria} 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 d05736c..db6600a 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 @@ -218,7 +218,7 @@ piccola di $\PP(\RR)$, la $\sigma$-algebra dei boreliani, che ci permette di esc \begin{proposition}[Proprietà della f.d.r.] Sia $P$ una probabilità reale. Allora, se $F$ è la - sua f.d.r. vale che: + sua f.d.r.~vale che: \begin{enumerate}[(i.)] \item $F$ è crescente, ovvero $F(x) \geq F(y) \impliedby x \geq y$ (infatti $(-\infty, x] \supseteq (-\infty, y]$), \item $F$ è continua a destra, ovverosia per ogni $\tilde{x} \in \RR$ vale che $\lim_{x \to \tilde{x}^+} F(x) = F(\tilde{x})$, @@ -329,7 +329,7 @@ si dividono dunque secondo il seguente schema: \] Allora $F$ è crescente, continua a destra e tale per cui $\lim_{x \to -\infty} F(x) = 0$, $\lim_{x \to \infty} F(x) = 1$. - Pertanto esiste un'unica probabilità $P$ avente $F$ come f.d.r. per la + Pertanto esiste un'unica probabilità $P$ avente $F$ come f.d.r.~per la \textit{Proposizione \ref{prop:unicita_fdr}}. \smallskip @@ -399,7 +399,7 @@ i risultati della \textit{Parte 2}. \end{definition} \begin{remark} - Se $P$ è AC, allora la sua f.d.r. $F$ è in particolare + Se $P$ è AC, allora la sua f.d.r.~$F$ è in particolare assolutamente continua, e dunque anche continua. \end{remark} @@ -432,7 +432,7 @@ i risultati della \textit{Parte 2}. \end{proposition} \begin{proposition}[Relazioni tra la densità e la f.d.r.] - Sia $P$ una probabilità reale con f.d.r. $F$. Allora valgono le seguenti affermazioni: + Sia $P$ una probabilità reale con f.d.r.~$F$. Allora valgono le seguenti affermazioni: \begin{enumerate}[(i.)] \item Se $P$ è AC con densità $f$, allora $F(x) = \int_{-\infty}^x f(y) \dy$. Viceversa se esiste $f$ per cui $F(x) = \int_{-\infty}^x f(y) \dy$, allora $P$ è AC con densità diff --git a/Secondo anno/Elementi di probabilità e statistica/sections/tabella-modelli-reali.tex b/Secondo anno/Elementi di probabilità e statistica/sections/tabella-modelli-reali.tex new file mode 100644 index 0000000..8ab94b4 --- /dev/null +++ b/Secondo anno/Elementi di probabilità e statistica/sections/tabella-modelli-reali.tex @@ -0,0 +1,34 @@ +%-------------------------------------------------------------------- +\begin{landscape} + + \chapter*{Tabella e proprietà delle distribuzioni assolutamente continue} + \addcontentsline{toc}{chapter}{Tabella e proprietà delle distribuzioni assolutamente continue} + + \vskip -0.3in + + \begin{center} + \begin{table}[htb] + \scalebox{1.1}{ + \begin{tabular}{|l|l|l|l|l|l|} + \hline + Nome distribuzione & Caso di utilizzo & Parametri & Densità & Funzione di ripartizione & Probabilità \\ \hline + \begin{tabular}[c]{@{}l@{}}Distr. uniforme\\ $X \sim U(B)$\end{tabular} & \begin{tabular}[c]{@{}l@{}}Estrazione di un\\ punto reale a caso\\ su $B$ senza preferenze.\end{tabular} & \begin{tabular}[c]{@{}l@{}}$B \in \BB(\RR)$ -- insieme\\ da cui estrarre.\end{tabular} & $f(x) = \frac{1}{m(B)} 1_B(x)$ & $F(x) = \frac{m((-\infty, x] \cap B)}{m(B)}$ & $P(X \in A) = \frac{m(A \cap B)}{m(B)}$ \\ \hline + \begin{tabular}[c]{@{}l@{}}Distr. esponenz.\\ $X \sim \Exp(\lambda)$\end{tabular} & \begin{tabular}[c]{@{}l@{}}Processo di Poisson\\ in senso continuo.\end{tabular} & \begin{tabular}[c]{@{}l@{}}$\lambda > 0$ -- parametro\\ di Poisson.\end{tabular} & $f(x) = \lambda e^{-\lambda x} 1_{(0, \infty)} (x)$ & \begin{tabular}[c]{@{}l@{}}$F(x) = 1-e^{-\lambda x}$\\ per $x \geq 0$, $0$ altrimenti.\end{tabular} & \\ \hline + \begin{tabular}[c]{@{}l@{}}Distr. gamma\\ $X \sim \Gamma(r, \lambda)$\end{tabular} & \begin{tabular}[c]{@{}l@{}}Estensione della distr.\\ binomiale in senso\\ continuo.\end{tabular} & $r > 0$, $\lambda > 0$. & $f(x) = \frac{\lambda^r}{\Gamma(r)} x^{r-1} e^{-\lambda x} 1_{(0, \infty)}(x)$ & & \\ \hline + \begin{tabular}[c]{@{}l@{}}Distr. normale\\ $X \sim N(\mu, \sigma^2)$\end{tabular} & & \begin{tabular}[c]{@{}l@{}}$\mu$ -- media.\\ $\sigma^2 > 0$ -- varianza.\end{tabular} & $f(x) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{-\nicefrac{(x-\mu)^2}{2\sigma^2}}$ & \begin{tabular}[c]{@{}l@{}}$\Phi(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-\nicefrac{t^2}{2}} \dt$\\ per $N(0, 1)$ e si standardizza\\ le altre distr. con il cambio\\ di var. $z = \nicefrac{(x-\mu)}{\sigma}$.\end{tabular} & \\ \hline + \end{tabular} + } + \end{table} + \end{center} + + Si ricorda che la funzione $\Gamma$ è definita in modo tale che $\Gamma(x) = \int_0^\infty t^{x-1} e^{-t} \dt$; si tratta + di un'estensione della nozione di fattoriale ai valori reali (infatti, $\Gamma(n+1) = n!$ per $n \in \NN$). + + % Valgono inoltre le seguenti altre proprietà: + % + % \small + % \begin{itemize} + % \item + % \end{itemize} + + \end{landscape} \ No newline at end of file diff --git a/Secondo anno/Elementi di probabilità e statistica/sections/tabella-phi.tex b/Secondo anno/Elementi di probabilità e statistica/sections/tabella-phi.tex index 79732cf..087a409 100644 --- a/Secondo anno/Elementi di probabilità e statistica/sections/tabella-phi.tex +++ b/Secondo anno/Elementi di probabilità e statistica/sections/tabella-phi.tex @@ -16,7 +16,7 @@ $a > 0$, allora $\int_{-a}^a e^{-\nicefrac{x^2}{2}} \dx = \Phi(a) - \Phi(-a) = 2 \endhead -\hline \multicolumn{3}{|r|}{{Continua nella prossima pagina}} \\ \hline +\hline \multicolumn{3}{|r|}{{Continua...}} \\ \hline \endfoot \hline