From 7a9e13a9f8777dc8d2a75197f03e98b5a1825a0e Mon Sep 17 00:00:00 2001 From: Hearot Date: Mon, 22 Apr 2024 20:22:49 +0200 Subject: [PATCH] =?UTF-8?q?feat(eps):=20probabilit=C3=A0=20assolutamente?= =?UTF-8?q?=20continue?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../main.pdf | 5213 +++++++++-------- .../preamble.tex | 5 +- .../sections/0-notazioni.tex | 3 + .../sections/3-probabilità-reale.tex | 103 +- 4 files changed, 2899 insertions(+), 2425 deletions(-) diff --git a/Secondo anno/Elementi di probabilità e statistica/main.pdf b/Secondo anno/Elementi di probabilità e statistica/main.pdf index 0c5f5cc..3e31a13 100644 --- a/Secondo anno/Elementi di probabilità e statistica/main.pdf +++ b/Secondo anno/Elementi di probabilità e statistica/main.pdf @@ -406,62 +406,80 @@ endobj << /S /GoTo /D (subsection.3.2.1) >> endobj 275 0 obj 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(chapter*.2) 5 0 R (chapter*.9) 33 0 R] +/Limits [(chapter*.1) (chapter*.9)] >> endobj -875 0 obj +912 0 obj << -/Names [(page.10) 501 0 R (page.11) 521 0 R (page.12) 529 0 R (page.13) 549 0 R (page.14) 569 0 R (page.15) 591 0 R] -/Limits [(page.10) (page.15)] +/Names [(chapter.1) 61 0 R (chapter.2) 125 0 R (chapter.3) 249 0 R (page.1) 314 0 R (page.10) 515 0 R (page.11) 535 0 R] +/Limits [(chapter.1) (page.11)] >> endobj -876 0 obj +913 0 obj << -/Names [(page.16) 614 0 R (page.17) 640 0 R (page.18) 663 0 R (page.19) 684 0 R (page.2) 418 0 R (page.20) 698 0 R] -/Limits [(page.16) (page.20)] +/Names [(page.12) 543 0 R (page.13) 563 0 R (page.14) 583 0 R (page.15) 605 0 R (page.16) 628 0 R (page.17) 654 0 R] +/Limits [(page.12) (page.17)] >> endobj -877 0 obj +914 0 obj << -/Names [(page.21) 712 0 R (page.22) 717 0 R (page.23) 737 0 R (page.24) 775 0 R (page.25) 787 0 R (page.26) 792 0 R] -/Limits [(page.21) (page.26)] +/Names [(page.18) 677 0 R (page.19) 698 0 R (page.2) 432 0 R (page.20) 712 0 R (page.21) 727 0 R (page.22) 732 0 R] +/Limits [(page.18) (page.22)] >> endobj -878 0 obj +915 0 obj << -/Names [(page.3) 424 0 R (page.4) 429 0 R (page.5) 433 0 R (page.6) 438 0 R (page.7) 443 0 R (page.8) 448 0 R] -/Limits [(page.3) (page.8)] +/Names [(page.23) 752 0 R (page.24) 786 0 R (page.25) 803 0 R (page.26) 820 0 R (page.27) 825 0 R (page.3) 438 0 R] +/Limits [(page.23) (page.3)] >> endobj -879 0 obj +916 0 obj << -/Names [(page.9) 467 0 R (section*.10) 37 0 R (section*.11) 41 0 R (section*.12) 45 0 R (section*.13) 49 0 R (section*.14) 53 0 R] -/Limits [(page.9) (section*.14)] +/Names [(page.4) 443 0 R (page.5) 447 0 R (page.6) 452 0 R (page.7) 457 0 R (page.8) 462 0 R (page.9) 481 0 R] +/Limits [(page.4) (page.9)] >> endobj -880 0 obj +917 0 obj << -/Names [(section*.16) 444 0 R (section*.3) 9 0 R (section*.4) 13 0 R (section*.5) 17 0 R (section*.6) 21 0 R (section*.7) 25 0 R] -/Limits [(section*.16) (section*.7)] +/Names [(section*.10) 37 0 R (section*.11) 41 0 R (section*.12) 45 0 R (section*.13) 49 0 R (section*.14) 53 0 R (section*.16) 458 0 R] +/Limits [(section*.10) (section*.16)] >> endobj -881 0 obj +918 0 obj << -/Names [(section*.8) 29 0 R (section.1.1) 65 0 R (section.1.2) 85 0 R (section.1.3) 89 0 R (section.1.4) 105 0 R (section.1.5) 121 0 R] -/Limits [(section*.8) (section.1.5)] +/Names [(section*.3) 9 0 R (section*.4) 13 0 R (section*.5) 17 0 R (section*.6) 21 0 R (section*.7) 25 0 R (section*.8) 29 0 R] +/Limits [(section*.3) (section*.8)] >> endobj -882 0 obj +919 0 obj << -/Names [(section.2.1) 129 0 R (section.2.2) 145 0 R (section.2.3) 169 0 R (section.2.4) 193 0 R (section.2.5) 197 0 R (section.2.6) 213 0 R] -/Limits [(section.2.1) (section.2.6)] +/Names [(section.1.1) 65 0 R (section.1.2) 85 0 R (section.1.3) 89 0 R (section.1.4) 105 0 R (section.1.5) 121 0 R (section.2.1) 129 0 R] +/Limits [(section.1.1) (section.2.1)] >> endobj -883 0 obj +920 0 obj << -/Names [(section.2.7) 225 0 R (section.2.8) 237 0 R (section.3.1) 253 0 R (section.3.2) 269 0 R (section.3.3) 277 0 R (subsection.1.1.1) 69 0 R] -/Limits [(section.2.7) (subsection.1.1.1)] +/Names [(section.2.2) 145 0 R (section.2.3) 169 0 R (section.2.4) 193 0 R (section.2.5) 197 0 R (section.2.6) 213 0 R (section.2.7) 225 0 R] +/Limits [(section.2.2) (section.2.7)] >> endobj -884 0 obj +921 0 obj << -/Names [(subsection.1.1.2) 73 0 R (subsection.1.1.3) 77 0 R (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] -/Limits [(subsection.1.1.2) (subsection.1.3.3)] +/Names [(section.2.8) 237 0 R (section.3.1) 253 0 R (section.3.2) 269 0 R (section.3.3) 281 0 R (section.3.4) 285 0 R (section.3.5) 289 0 R] +/Limits [(section.2.8) (section.3.5)] >> endobj -885 0 obj +922 0 obj << -/Names [(subsection.1.4.1) 109 0 R (subsection.1.4.2) 113 0 R (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] -/Limits [(subsection.1.4.1) (subsection.2.1.3)] +/Names [(subsection.1.1.1) 69 0 R (subsection.1.1.2) 73 0 R (subsection.1.1.3) 77 0 R (subsection.1.1.4) 81 0 R (subsection.1.3.1) 93 0 R (subsection.1.3.2) 97 0 R] +/Limits [(subsection.1.1.1) (subsection.1.3.2)] >> endobj -886 0 obj +923 0 obj << -/Names [(subsection.2.2.1) 149 0 R (subsection.2.2.2) 153 0 R (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] -/Limits [(subsection.2.2.1) (subsection.2.3.1)] +/Names [(subsection.1.3.3) 101 0 R (subsection.1.4.1) 109 0 R (subsection.1.4.2) 113 0 R (subsection.1.4.3) 117 0 R (subsection.2.1.1) 133 0 R (subsection.2.1.2) 137 0 R] +/Limits [(subsection.1.3.3) (subsection.2.1.2)] >> endobj -887 0 obj +924 0 obj << -/Names [(subsection.2.3.2) 177 0 R (subsection.2.3.3) 181 0 R (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] -/Limits [(subsection.2.3.2) (subsection.2.5.2)] +/Names [(subsection.2.1.3) 141 0 R (subsection.2.2.1) 149 0 R (subsection.2.2.2) 153 0 R (subsection.2.2.3) 157 0 R (subsection.2.2.4) 161 0 R (subsection.2.2.5) 165 0 R] +/Limits [(subsection.2.1.3) (subsection.2.2.5)] >> endobj -888 0 obj +925 0 obj << -/Names [(subsection.2.5.3) 209 0 R (subsection.2.6.1) 217 0 R (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] -/Limits [(subsection.2.5.3) (subsection.2.8.1)] +/Names [(subsection.2.3.1) 173 0 R (subsection.2.3.2) 177 0 R (subsection.2.3.3) 181 0 R (subsection.2.3.4) 185 0 R (subsection.2.3.5) 189 0 R (subsection.2.5.1) 201 0 R] +/Limits [(subsection.2.3.1) (subsection.2.5.1)] >> endobj -889 0 obj +926 0 obj << -/Names [(subsection.2.8.2) 245 0 R (subsection.3.1.1) 257 0 R (subsection.3.1.2) 261 0 R (subsection.3.1.3) 265 0 R (subsection.3.2.1) 273 0 R (subsection.3.3.1) 281 0 R] -/Limits [(subsection.2.8.2) (subsection.3.3.1)] +/Names [(subsection.2.5.2) 205 0 R (subsection.2.5.3) 209 0 R (subsection.2.6.1) 217 0 R (subsection.2.6.2) 221 0 R (subsection.2.7.1) 229 0 R (subsection.2.7.2) 233 0 R] +/Limits [(subsection.2.5.2) (subsection.2.7.2)] >> endobj -890 0 obj +927 0 obj << -/Names [(subsection.3.3.2) 285 0 R (table.3.1) 708 0 R (theorem.1.1) 449 0 R (theorem.1.10) 468 0 R (theorem.1.11) 472 0 R (theorem.1.12) 473 0 R] -/Limits [(subsection.3.3.2) (theorem.1.12)] +/Names [(subsection.2.8.1) 241 0 R (subsection.2.8.2) 245 0 R (subsection.3.1.1) 257 0 R (subsection.3.1.2) 261 0 R (subsection.3.1.3) 265 0 R (subsection.3.2.1) 273 0 R] +/Limits [(subsection.2.8.1) (subsection.3.2.1)] >> endobj -891 0 obj +928 0 obj << -/Names [(theorem.1.13) 482 0 R (theorem.1.14) 483 0 R (theorem.1.15) 486 0 R (theorem.1.16) 487 0 R (theorem.1.17) 488 0 R (theorem.1.18) 489 0 R] -/Limits [(theorem.1.13) (theorem.1.18)] +/Names [(subsection.3.2.2) 277 0 R (subsection.3.5.1) 293 0 R (subsection.3.5.2) 297 0 R (table.3.1) 723 0 R (theorem.1.1) 463 0 R (theorem.1.10) 482 0 R] +/Limits [(subsection.3.2.2) (theorem.1.10)] >> endobj -892 0 obj +929 0 obj << -/Names [(theorem.1.19) 490 0 R (theorem.1.2) 450 0 R (theorem.1.20) 491 0 R (theorem.1.21) 492 0 R (theorem.1.22) 493 0 R (theorem.1.23) 494 0 R] -/Limits [(theorem.1.19) (theorem.1.23)] +/Names [(theorem.1.11) 486 0 R (theorem.1.12) 487 0 R (theorem.1.13) 496 0 R (theorem.1.14) 497 0 R (theorem.1.15) 500 0 R (theorem.1.16) 501 0 R] +/Limits [(theorem.1.11) (theorem.1.16)] >> endobj -893 0 obj +930 0 obj << -/Names [(theorem.1.24) 495 0 R (theorem.1.25) 496 0 R (theorem.1.26) 502 0 R (theorem.1.27) 503 0 R (theorem.1.28) 504 0 R (theorem.1.29) 505 0 R] -/Limits [(theorem.1.24) (theorem.1.29)] +/Names [(theorem.1.17) 502 0 R (theorem.1.18) 503 0 R (theorem.1.19) 504 0 R (theorem.1.2) 464 0 R (theorem.1.20) 505 0 R (theorem.1.21) 506 0 R] +/Limits [(theorem.1.17) (theorem.1.21)] >> endobj -894 0 obj +931 0 obj << -/Names [(theorem.1.3) 451 0 R (theorem.1.30) 506 0 R (theorem.1.31) 507 0 R (theorem.1.32) 508 0 R (theorem.1.33) 509 0 R (theorem.1.34) 511 0 R] -/Limits [(theorem.1.3) (theorem.1.34)] +/Names [(theorem.1.22) 507 0 R (theorem.1.23) 508 0 R (theorem.1.24) 509 0 R (theorem.1.25) 510 0 R (theorem.1.26) 516 0 R (theorem.1.27) 517 0 R] +/Limits [(theorem.1.22) (theorem.1.27)] >> endobj -895 0 obj +932 0 obj << -/Names [(theorem.1.35) 512 0 R (theorem.1.36) 513 0 R (theorem.1.37) 514 0 R (theorem.1.38) 515 0 R (theorem.1.39) 516 0 R (theorem.1.4) 455 0 R] -/Limits [(theorem.1.35) (theorem.1.4)] +/Names [(theorem.1.28) 518 0 R (theorem.1.29) 519 0 R (theorem.1.3) 465 0 R (theorem.1.30) 520 0 R (theorem.1.31) 521 0 R (theorem.1.32) 522 0 R] +/Limits [(theorem.1.28) (theorem.1.32)] >> endobj -896 0 obj +933 0 obj << -/Names [(theorem.1.40) 517 0 R (theorem.1.41) 522 0 R (theorem.1.42) 523 0 R (theorem.1.43) 524 0 R (theorem.1.44) 525 0 R (theorem.1.5) 456 0 R] -/Limits [(theorem.1.40) (theorem.1.5)] +/Names [(theorem.1.33) 523 0 R (theorem.1.34) 525 0 R (theorem.1.35) 526 0 R (theorem.1.36) 527 0 R (theorem.1.37) 528 0 R (theorem.1.38) 529 0 R] +/Limits [(theorem.1.33) (theorem.1.38)] >> endobj -897 0 obj +934 0 obj << -/Names [(theorem.1.6) 457 0 R (theorem.1.7) 458 0 R (theorem.1.8) 459 0 R (theorem.1.9) 463 0 R (theorem.2.1) 530 0 R (theorem.2.10) 541 0 R] -/Limits [(theorem.1.6) (theorem.2.10)] +/Names [(theorem.1.39) 530 0 R (theorem.1.4) 469 0 R (theorem.1.40) 531 0 R (theorem.1.41) 536 0 R (theorem.1.42) 537 0 R (theorem.1.43) 538 0 R] +/Limits [(theorem.1.39) (theorem.1.43)] >> endobj -898 0 obj +935 0 obj << -/Names [(theorem.2.100) 690 0 R (theorem.2.101) 691 0 R (theorem.2.102) 699 0 R (theorem.2.103) 701 0 R (theorem.2.104) 702 0 R (theorem.2.105) 703 0 R] -/Limits [(theorem.2.100) (theorem.2.105)] +/Names [(theorem.1.44) 539 0 R (theorem.1.5) 470 0 R (theorem.1.6) 471 0 R (theorem.1.7) 472 0 R (theorem.1.8) 473 0 R (theorem.1.9) 477 0 R] +/Limits [(theorem.1.44) (theorem.1.9)] >> endobj -899 0 obj +936 0 obj << -/Names [(theorem.2.106) 704 0 R (theorem.2.107) 705 0 R (theorem.2.108) 706 0 R (theorem.2.11) 542 0 R (theorem.2.12) 543 0 R (theorem.2.13) 550 0 R] -/Limits [(theorem.2.106) (theorem.2.13)] +/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)] >> endobj -900 0 obj +937 0 obj << -/Names [(theorem.2.14) 551 0 R (theorem.2.15) 552 0 R (theorem.2.16) 553 0 R (theorem.2.17) 554 0 R (theorem.2.18) 555 0 R (theorem.2.19) 556 0 R] -/Limits [(theorem.2.14) (theorem.2.19)] +/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)] >> endobj -901 0 obj +938 0 obj << -/Names [(theorem.2.2) 531 0 R (theorem.2.20) 557 0 R (theorem.2.21) 558 0 R (theorem.2.22) 559 0 R (theorem.2.23) 560 0 R (theorem.2.24) 561 0 R] -/Limits [(theorem.2.2) (theorem.2.24)] +/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)] >> endobj -902 0 obj +939 0 obj << -/Names [(theorem.2.25) 562 0 R (theorem.2.26) 563 0 R (theorem.2.27) 570 0 R (theorem.2.28) 571 0 R (theorem.2.29) 572 0 R (theorem.2.3) 534 0 R] -/Limits [(theorem.2.25) (theorem.2.3)] +/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)] >> endobj -903 0 obj +940 0 obj << -/Names [(theorem.2.30) 573 0 R (theorem.2.31) 574 0 R (theorem.2.32) 575 0 R (theorem.2.33) 576 0 R (theorem.2.34) 577 0 R (theorem.2.35) 578 0 R] -/Limits [(theorem.2.30) (theorem.2.35)] +/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)] >> endobj -904 0 obj +941 0 obj << -/Names [(theorem.2.36) 579 0 R (theorem.2.37) 580 0 R (theorem.2.38) 582 0 R (theorem.2.39) 583 0 R (theorem.2.4) 535 0 R (theorem.2.40) 584 0 R] -/Limits [(theorem.2.36) (theorem.2.40)] +/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)] >> endobj -905 0 obj +942 0 obj << -/Names [(theorem.2.41) 585 0 R (theorem.2.42) 592 0 R (theorem.2.43) 593 0 R (theorem.2.44) 594 0 R (theorem.2.45) 595 0 R (theorem.2.46) 598 0 R] -/Limits [(theorem.2.41) (theorem.2.46)] +/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)] >> endobj -906 0 obj +943 0 obj << -/Names [(theorem.2.47) 599 0 R (theorem.2.48) 600 0 R (theorem.2.49) 601 0 R (theorem.2.5) 536 0 R (theorem.2.50) 602 0 R (theorem.2.51) 603 0 R] -/Limits [(theorem.2.47) (theorem.2.51)] +/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)] >> endobj -907 0 obj +944 0 obj << -/Names [(theorem.2.52) 604 0 R (theorem.2.53) 607 0 R (theorem.2.54) 615 0 R (theorem.2.55) 622 0 R (theorem.2.56) 623 0 R (theorem.2.57) 624 0 R] -/Limits [(theorem.2.52) (theorem.2.57)] +/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)] >> endobj -908 0 obj +945 0 obj << -/Names [(theorem.2.58) 625 0 R (theorem.2.59) 626 0 R (theorem.2.6) 537 0 R (theorem.2.60) 627 0 R (theorem.2.61) 628 0 R (theorem.2.62) 629 0 R] -/Limits [(theorem.2.58) (theorem.2.62)] +/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) 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-\newcommand{\dx}{\mathop{dx}} +\newcommand*\dif{\mathop{}\!\textnormal{\slshape d}} +\newcommand{\dx}{\dif{x}} +\newcommand{\dy}{\dif{dy}} + %\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 ebea5bd..19a75d2 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 @@ -66,6 +66,8 @@ modo $f = f^+ - f^-$. \item $\exp$ -- Funzione esponenziale $e^x$. \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. \end{itemize} \section*{Combinatoria} @@ -166,6 +168,7 @@ discreta). \item f.d.r.~-- funzione di ripartizione, rispetto a una probabilità reale. \item $F$, $F_P$ -- per una probabilità reale, funzione di ripartizione. + \item $f$ -- densità (in senso reale) della probabilità. \item AC -- assolutamente continua, riferito a una probabilità. \item \va -- variabile aleatoria. \item $P^X$ -- legge della v.a.~$X$ rispetto a $P$. 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 888c86e..d05736c 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 @@ -197,7 +197,7 @@ piccola di $\PP(\RR)$, la $\sigma$-algebra dei boreliani, che ci permette di esc \section{Probabilità reale, funzione di ripartizione (f.d.r.) e proprietà} -\subsection{Definizioni e corrispondenza tra f.d.r.~e probabilità reale} +\subsection{Definizioni e proprietà della f.d.r.} \begin{definition} Si dice \textbf{probabilità reale} una qualsiasi @@ -233,7 +233,10 @@ piccola di $\PP(\RR)$, la $\sigma$-algebra dei boreliani, che ci permette di esc $(F(x_i))_{i \in \NN} = (P((-\infty, x_i]))_{i \in \NN} \goesdown P((-\infty, \tilde{x}]) = F(\tilde{x})$. \end{proposition} +\subsection{Corrispondenza tra f.d.r.~e probabilità, calcolo di \texorpdfstring{$P$}{P} tramite \texorpdfstring{$F$}{F} e probabilità continue} + \begin{proposition}[$P$ è univocamente determinata da $F$] + \label{prop:unicita_fdr} Sia $F : \RR \to \RR$ una funzione tale per cui: \begin{enumerate}[(i.)] \item $F$ è crescente, @@ -284,7 +287,7 @@ piccola di $\PP(\RR)$, la $\sigma$-algebra dei boreliani, che ci permette di esc a $F(b) - F(a)$ in tutti i casi (infatti $F(a^-) = F(a)$ e $F(b^-) = F(b)$). \end{remark} -\section{Classi di probabilità reale: discreta e assolutamente continua (AC)} +\section{Classi principali di probabilità reale} Esistono due classi importanti, ma non esaustive, di probabilità reale: le probabilità discrete e quelle assolutamente @@ -315,7 +318,28 @@ si dividono dunque secondo il seguente schema: \end{tikzpicture} \end{center} -\subsection{Probabilità discreta e f.d.r.} +\begin{example}[Esempio di probabilità né discreta né continua] + Sia $F : \RR \to \RR$ tale per cui: + \[ + F(x) = \begin{cases} + 0 & \text{se } x < 0, \\ + x + \frac{1}{2} & \text{se } 0 \leq x \leq \frac{1}{2}, \\ + 1 & \text{altrimenti}. + \end{cases} + \] + 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 + \textit{Proposizione \ref{prop:unicita_fdr}}. \smallskip + + + Dal momento che $F$ non è continua a sinistra in $0$, $F$ non è continua, e dunque + $P$ non è continua. Inoltre $F$ non induce una probabilità discreta dacché + non è costante a tratti in $[0, \nicefrac{1}{2}]$. Pertanto $P$ non è né + continua né discreta. +\end{example} + +\section{Probabilità discreta e rappresentazione della f.d.r.} Come già discusso nella sezione della \textit{\hyperref[sec:discretizzazione]{Discretizzazione}}, una probabilità reale $P$ si dice \textit{discreta} se esiste $\Omega_0 \subseteq \RR$ @@ -350,12 +374,81 @@ In questo caso il range $R_P$ è dunque numerabile e, se $F$ è la f.d.r.~di $P$ Pertanto, se una probabilità reale è discreta, ci si può effettivamente restringere a tutti i risultati della \textit{Parte 2}. -\subsection{Probabilità assolutamente continue (AC)} +\section{Probabilità assolutamente continue (AC)} \begin{warn} Si ricorda che con il simbolo $\int$ si intende l'integrale secondo Lebesgue e che si assume di star lavorando sempre con la - misura di Lebesgue. + misura di Lebesgue $m$. \end{warn} +\subsection{Probabilità AC e funzione di densità} + +\begin{definition}[Probabilità assolutamente continua (AC) e densità] + Una probabilità $P$ si dice \textbf{assolutamente continua (AC)} + se esiste una funzione + boreliana $f : \RR \to \RR$ tale per cui: + \[ + P(A) = \int_A f(x) \dx, + \] + dove si impiega l'integrale secondo Lebesgue. Tale funzione $f$ è + detta \textbf{densità} di $P$. \smallskip + + + Si assume implicitamente che $\int_\RR \abs{f(x)} \dx$ sia finito. +\end{definition} + +\begin{remark} + Se $P$ è AC, allora la sua f.d.r. $F$ è in particolare + assolutamente continua, e dunque anche continua. +\end{remark} + +\begin{remark} + Nella pratica l'integrale $\int_A f(x) \dx$ si riduce in molti casi + al più semplice integrale di Riemann, eventualmente improprio. +\end{remark} + +\subsection{Proprietà e caratterizzazione della densità} + +\begin{proposition}[Unicità della densità a meno di $m$-trascurabilità] + Se $P$ è una probabilità AC con densità $f$ e $g$, allora + $f = g$ q.o.~(e dunque $m(f \neq g) = 0$, ossia l'insieme + $f \neq g$ è $m$-trascurabile). +\end{proposition} + +\begin{remark} + Si osserva che se $P$ è una probabilità AC con densità + $f$, allora $f \geq 0$ q.o.~per continuità (altrimenti $P$ potrebbe + assumere valori negativi) e $\int_\RR f(x) \dx = P(\RR) = 1$. +\end{remark} + +\begin{proposition}[La densità determina univocamente la probabilità] + Sia $f : \RR \to \RR$ una funzione boreliana tale per cui: + \begin{enumerate}[(i.)] + \item $f \geq 0$, + \item $\int_\RR f(x) \dx = 1$. + \end{enumerate} + Allora esiste un'unica probabilità reale $P$ avente $f$ come densità. +\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: + \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 + $f$. + \item Se $F$ è continua e $C^1$ a tratti (ovverosia si restringe a una funzione $C^1$ eccetto che per un insieme di punti isolati), + allora $P$ è AC con densità $f$ t.c.~$f = F'$ dove è definibile $F'$ e $f = 0$ altrimenti (segue dal Teorema fondamentale del calcolo integrale). + \end{enumerate} +\end{proposition} + +\begin{remark} + Se $P$ è AC con densità $f$, allora $P(f = 0) = \int_{f = 0} f(x) \dx = 0$ e dunque + l'insieme $f = 0$ è trascurabile rispetto a $P$. Dunque, ristringendo il range si + ottiene che: + \[ + P(A) = P(A \cap (f > 0)). + \] +\end{remark} + \end{multicols*} \ No newline at end of file