From 256ad2942c5f139621cbf2972eb3fda747f1a1d1 Mon Sep 17 00:00:00 2001 From: Hearot Date: Mon, 22 Apr 2024 22:42:15 +0200 Subject: [PATCH] fix(eps): aggiunge la costante 1/sqrt(2pi) --- .../main.pdf | 1812 +++++++++-------- .../sections/2-probabilità-discreta.tex | 3 +- .../sections/tabella-modelli-discreti.tex | 1 + .../sections/tabella-phi.tex | 6 +- 4 files changed, 919 insertions(+), 903 deletions(-) diff --git a/Secondo anno/Elementi di probabilità e statistica/main.pdf b/Secondo anno/Elementi di probabilità e statistica/main.pdf index 81410e5..45bfd2d 100644 --- a/Secondo anno/Elementi di probabilità e statistica/main.pdf +++ b/Secondo anno/Elementi di probabilità e statistica/main.pdf @@ -3842,7 +3842,7 @@ endobj /ProcSet [ /PDF /Text ] >> endobj -722 0 obj +723 0 obj << /Length 8557 /Filter /FlateDecode @@ -3892,11 +3892,11 @@ M w5L̼f^:zZ@x[%azdwg_ V==Gc|Иqp'-έOsp8rX΋ˤG264X)l"{˷Rk.PTI}*@Wh,90fv~q1^YUv`ԃ.S+Ұ)$ȚL(B EXG^?Ig endstream endobj -721 0 obj +722 0 obj << /Type /Page 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(section.2.2) 145 0 R (section.2.3) 169 0 R (section.2.4) 193 0 R] /Limits [(section.1.4) (section.2.4)] >> endobj -935 0 obj +936 0 obj << /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 +937 0 obj << /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 +938 0 obj << /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 +939 0 obj << /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 +940 0 obj << /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 +941 0 obj << /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 +942 0 obj << /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 +943 0 obj << /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 +944 0 obj << -/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] +/Names [(table.3.1) 734 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 +945 0 obj << /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 +946 0 obj << /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 +947 0 obj << /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 +948 0 obj << /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 +949 0 obj << /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 +950 0 obj << /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 +951 0 obj << /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 +952 0 obj << -/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] +/Names [(theorem.2.101) 716 0 R (theorem.2.102) 725 0 R (theorem.2.103) 727 0 R (theorem.2.104) 728 0 R (theorem.2.105) 729 0 R (theorem.2.106) 730 0 R] /Limits [(theorem.2.101) (theorem.2.106)] >> endobj -952 0 obj +953 0 obj << -/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] +/Names [(theorem.2.107) 731 0 R (theorem.2.108) 732 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 +954 0 obj << /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 +955 0 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(theorem.2.46) 623 0 R (theorem.2.47) 624 0 R] /Limits [(theorem.2.42) (theorem.2.47)] >> endobj -959 0 obj +960 0 obj << /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 +961 0 obj << /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 +962 0 obj << /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 +963 0 obj << /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 +964 0 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(theorem.2.95) 710 0 R (theorem.2.96) 711 0 R] /Limits [(theorem.2.91) (theorem.2.96)] >> endobj -968 0 obj +969 0 obj << -/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] +/Names [(theorem.2.97) 712 0 R (theorem.2.98) 713 0 R (theorem.2.99) 714 0 R (theorem.3.1) 744 0 R (theorem.3.10) 768 0 R (theorem.3.11) 775 0 R] /Limits [(theorem.2.97) (theorem.3.11)] >> endobj -969 0 obj +970 0 obj << -/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] +/Names [(theorem.3.12) 776 0 R (theorem.3.13) 779 0 R (theorem.3.14) 780 0 R (theorem.3.15) 781 0 R (theorem.3.16) 782 0 R (theorem.3.17) 783 0 R] /Limits [(theorem.3.12) (theorem.3.17)] >> endobj -970 0 obj +971 0 obj << -/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] +/Names [(theorem.3.18) 784 0 R (theorem.3.19) 785 0 R (theorem.3.2) 745 0 R (theorem.3.20) 786 0 R (theorem.3.21) 799 0 R (theorem.3.22) 804 0 R] /Limits [(theorem.3.18) (theorem.3.22)] >> endobj -971 0 obj +972 0 obj << -/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] +/Names [(theorem.3.23) 805 0 R (theorem.3.24) 806 0 R (theorem.3.25) 807 0 R (theorem.3.26) 808 0 R (theorem.3.27) 809 0 R (theorem.3.28) 810 0 R] /Limits [(theorem.3.23) (theorem.3.28)] >> endobj -972 0 obj +973 0 obj << -/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 (theorem.3.33) 818 0 R] +/Names [(theorem.3.29) 815 0 R (theorem.3.3) 749 0 R (theorem.3.30) 816 0 R (theorem.3.31) 817 0 R (theorem.3.32) 818 0 R (theorem.3.33) 819 0 R] /Limits [(theorem.3.29) (theorem.3.33)] >> endobj -973 0 obj +974 0 obj << -/Names [(theorem.3.34) 819 0 R (theorem.3.35) 822 0 R (theorem.3.36) 825 0 R (theorem.3.4) 754 0 R (theorem.3.5) 755 0 R (theorem.3.6) 756 0 R] +/Names [(theorem.3.34) 820 0 R (theorem.3.35) 823 0 R (theorem.3.36) 826 0 R (theorem.3.4) 755 0 R (theorem.3.5) 756 0 R (theorem.3.6) 757 0 R] /Limits [(theorem.3.34) (theorem.3.6)] >> endobj -974 0 obj +975 0 obj << -/Names [(theorem.3.7) 757 0 R (theorem.3.8) 763 0 R (theorem.3.9) 766 0 R] +/Names [(theorem.3.7) 758 0 R (theorem.3.8) 764 0 R (theorem.3.9) 767 0 R] /Limits [(theorem.3.7) (theorem.3.9)] >> endobj -975 0 obj +976 0 obj << -/Kids [913 0 R 914 0 R 915 0 R 916 0 R 917 0 R 918 0 R] +/Kids [914 0 R 915 0 R 916 0 R 917 0 R 918 0 R 919 0 R] /Limits [(Doc-Start) (Item.33)] >> endobj -976 0 obj +977 0 obj << -/Kids [919 0 R 920 0 R 921 0 R 922 0 R 923 0 R 924 0 R] +/Kids [920 0 R 921 0 R 922 0 R 923 0 R 924 0 R 925 0 R] /Limits [(Item.34) (Item.9)] >> endobj -977 0 obj +978 0 obj << -/Kids [925 0 R 926 0 R 927 0 R 928 0 R 929 0 R 930 0 R] +/Kids [926 0 R 927 0 R 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+0000391198 00000 n trailer -<< /Size 992 -/Root 990 0 R -/Info 991 0 R -/ID [ ] >> +<< /Size 993 +/Root 991 0 R +/Info 992 0 R +/ID [<9113F774D9EA452D0124108608C31818> <9113F774D9EA452D0124108608C31818>] >> startxref -391062 +391472 %%EOF 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 327e90d..9a1ae29 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 @@ -1222,5 +1222,6 @@ di diverso, da cui l'intuizione del TCL. \end{definition} A partire dal modello delle prove ripetute si possono formalizzare - numerose distribuzioni, come quelle della sezione successiva. + numerose distribuzioni, come quelle della sezione delle + \textit{\hyperref[tab:distr_discrete]{Distribuzioni discrete}}. \end{multicols*} \ No newline at end of file diff --git a/Secondo anno/Elementi di probabilità e statistica/sections/tabella-modelli-discreti.tex b/Secondo anno/Elementi di probabilità e statistica/sections/tabella-modelli-discreti.tex index 299cd32..4877056 100644 --- a/Secondo anno/Elementi di probabilità e statistica/sections/tabella-modelli-discreti.tex +++ b/Secondo anno/Elementi di probabilità e statistica/sections/tabella-modelli-discreti.tex @@ -7,6 +7,7 @@ \vskip -0.3in \begin{table}[htb] +\label{tab:distr_discrete} \scalebox{0.74}{ \begin{tabular}{|l|l|l|l|l|l|l|} \hline 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 087a409..441fcf3 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 @@ -2,10 +2,10 @@ \chapter*{Tabella e proprietà della f.d.r.~\texorpdfstring{$\Phi(x)$}{Φ(x)} della normale standard \texorpdfstring{$N(0, 1)$}{N(0, 1)}} \addcontentsline{toc}{chapter}{Tabella e proprietà della f.d.r.~\texorpdfstring{$\Phi(x)$}{Φ(x)} della normale standard \texorpdfstring{$N(0, 1)$}{N(0, 1)}} -Per $a \in \RR$ si definisce la funzione $\Phi(a) = \int_{-\infty}^a e^{-\nicefrac{x^2}{2}} \dx$. L'integrale $\Phi(\infty) \defeq \int_{-\infty}^a e^{-\nicefrac{x^2}{2}} \dx$ vale +Per $a \in \RR$ si definisce la funzione $\Phi(a) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^a e^{-\nicefrac{x^2}{2}} \dx$. L'integrale $\Phi(\infty) \defeq \frac{1}{\sqrt{2\pi}} \int_{-\infty}^a e^{-\nicefrac{x^2}{2}} \dx$ vale esattamente $1$, mentre $\Phi(-\infty) \defeq 0$. Per la parità di $e^{-\nicefrac{x^2}{2}}$ vale che $\Phi(a) = 1 - \Phi(-a)$ (\textbf{simmetria}). A partire da questa funzione si può calcolare -$\int_{a}^b e^{-\nicefrac{x^2}{2}} \dx$, che risulta essere $\Phi(b) - \Phi(a)$. Se -$a > 0$, allora $\int_{-a}^a e^{-\nicefrac{x^2}{2}} \dx = \Phi(a) - \Phi(-a) = 2\Phi(a) - 1$. +$\frac{1}{\sqrt{2\pi}} \int_{a}^b e^{-\nicefrac{x^2}{2}} \dx$, che risulta essere $\Phi(b) - \Phi(a)$. Se +$a > 0$, allora $\frac{1}{\sqrt{2\pi}} \int_{-a}^a e^{-\nicefrac{x^2}{2}} \dx = \Phi(a) - \Phi(-a) = 2\Phi(a) - 1$. \begin{longtable}{|l|l|l|l|l|l|l|l|l|l|l|}