From 3d4987f9bf4782a33d0f3e7683e71638f7050b1d Mon Sep 17 00:00:00 2001 From: NotXia <35894453+NotXia@users.noreply.github.com> Date: Mon, 9 Oct 2023 21:48:55 +0200 Subject: [PATCH] Add SMM probability and random variables --- src/ainotes.cls | 1 + .../img/normal_distribution.png | Bin 0 -> 39959 bytes .../main.tex | 1 + .../sections/_probability.tex | 210 ++++++++++++++++++ 4 files changed, 212 insertions(+) create mode 100644 src/statistical-and-mathematical-methods-for-ai/img/normal_distribution.png create mode 100644 src/statistical-and-mathematical-methods-for-ai/sections/_probability.tex diff --git a/src/ainotes.cls b/src/ainotes.cls index 5312088..4370c4d 100644 --- a/src/ainotes.cls +++ b/src/ainotes.cls @@ -65,6 +65,7 @@ \renewcommand{\vec}[1]{{\mathbf{#1}}} \newcommand{\nullvec}[0]{\bar{\vec{0}}} \newcommand{\matr}[1]{{\bm{#1}}} +\newcommand{\prob}[1]{{\mathcal{P}({#1})}} \renewcommand*{\maketitle}{% diff --git a/src/statistical-and-mathematical-methods-for-ai/img/normal_distribution.png b/src/statistical-and-mathematical-methods-for-ai/img/normal_distribution.png new file mode 100644 index 0000000000000000000000000000000000000000..17d59b5c340865c848aa6aa987d5cbad622706cd GIT binary patch literal 39959 zcmb5W2Q-{v*ETwg-b?fro#>)P4-!O+kRTYnMUOVRi5@K?h(1X~qW2PY^dLg?ZnWr( zUPd_&dB6Yr&N=Hp|2oTBn8#D@dq4Z$*S@a3Z_!V+Rf!4c2|yqavAUYdGY|-K5Cp>b zg@*$ixhTw80DfTEJ=A�+lBaURYoQpYK|!J<|e#{J21%kT4MF95@uR1_F5rgFqYR zAdvJ&5QxStvtCyg_ydlmhN=qa`sNGX^eqWEg6pZS^$2$X4gDytv`;hWS{860c43CoP&K^Aqg^P#~Ojz-o z-{Jj?1EyE$6@0-X}59NsmRQB9Uxb)!Vx!7QRQtCSd>P zb9n6Cu=`qFRc(M6v`E z^!x{eNM$(fDV~(Fve4~C>2E@s@jmXh6ll02<&m$=0!VSM2pB&v(A*9`O%RR?QpF%0 zF;DA*DS#_Oybufsrl5v@L!}dd4}h8r7DQDsOCL?yo^St~@<^1Yc7Su?kLyiX4I)p` z_`W&SV8vUn`<`oqE?(tdN;bVc8SQNJ9(*4$J2JS`+P%UQOP1e+{&r#xzStxEt#EyD zIcml>wz!aq>E=!3c^c&17{Jxgbh@o$uvej2u5?nhJFIFNA^ z;f<-I(%&b4x+KsRzFC{EvJ7AK%V@pP+g7|NBKIKDO?`dj3B4f+b1)_DoUjC3VGd z)=5#=&0|%P2BBXm26^3l{&}~k&Jr)q%YVbbxBur&9RKl9z?E+7pk{T9J%=R$+Qp3Q z;rNg1S#HXb$YO61T`Ynx&ZUKFSdABr!6G%bP8nOt3i&85po40o@abl8vN80yHN_^0$dduWqmKipGS&c1Ep z8Df(xHt%)tMYox^z*9UExUw;~Zx;Sd!7{OsCJMe_O zVP?m%aUL^gK~ALZ_w^PI)7-zwe)lJtcpFmE3wAEtD~L_HI5FQs1zb0ofsfDFT7vaL z&}=yZNAo&`)<1L+W`$$}aTck@As!*&uo|I@lNZvBwK@E0iZ?HayjOPck6T!Q>|kEX z^we#i)iY=QhQ=4_)4Esa%=9gl z@gN_QxKF0{7(bK`oyuuInih=zO*fZbq;c^s*|8#nxdQyoX&Fha!C&&(JNDDamRc7? zQ0gq>uhQkp8gf>q+~(}R<6ljuKua)7bCtLhFx{!#+sXOPowi#?xa+^r z2`=sEn|u1$0FGTYL1UTrIE}~DxX1473f=86QTiKkjk2r`#%?TDHu?7=y)>K`b7~{S zulynK;_&M$EAVl{ugV{eoH@ziK8=GRXgwP59^Ai+Us+_yT`RtP2j6Ma6|bS67ZFVY zRwdDA{wrS_Vr)<6(lnN(a5c}|w)W|s;OjCB^ra}|YLzmi8*>}?4AYM@xm$HSHDvYdxGso0O3lYw?Z--leZ0hU$(UymAI}e-V96(Ns$7afB&GntHrh9eAPA3%#T7b z6~m{w{y}%g0)d7-Ccscg5AKSS_J~4 z!uD;sHCH<Ow0!* zIy?fW#jI}DY801dY7jdq#;R9i_sAc|&DODMSI_NWbZdov(kJn`+)6L#+;1t!Li{SK zu^eZBPp+Ay_6M~T^?>m?2CC#*I{s}DrA^(@%rzC_hdy-9A2j~-Sujj!nvjw>MKtHNCUDu+AQRLMWa!4u;F89_{65(#P7h6l@L z$FDjN4<){}ysH(yG0|CO6Q#sI2$?J_!=Ki_@fwwFb=vzl69N;|J~MWU@o3HE!UigC zN-OJfu7^gaU1;7qS1UZ79m)!aFQ3%Il<@vwtxF)|zZ9>`m}iI_$IHtEOEa5~)%pwK zdgz_mxj{~nb{T6AS2Aq-{Irx5gWtVDd*?-{N@!6n!)^=|Rh@Tt=y0ckn6ozJva<1P zr=>eRyj7?!9<6F5xb)y|ICd4;?UJXD6a??cQb@^jd}s2ft0Z{R$G}CR{xWhtB2`Rb zcj`L6-MTe4j}#KS6>LKQ06vfpe8@T7UP;U|Fwkr|D_%Q!q1oW@n_8(iz0dj~YuN8E zKp0LZV$yE;5?6h}=E*7?JGFt?b-{U|pe#*F`{N5g?+xq&6Xf`L&pRmGz#IeQcwQaI!Qg5yI>J+r9k+`k9M%LhTUWUQF*V2Br};W<_!;oKNcIX@$tA7my<>^Y|~Zb zlC5kJHbxuFESomIicq?>GuPl=u87Q!F=k39kGQO|;+2Xgz7jK*@y}@^$yWZKK=6}n zQM-qiEA#rgY2aI&VSVvRN~7rFea-KprZaVisgJ@z$Y)3lb4nCZS%W6q4Y08(mrV?} z5qm%LN);De*p>b(W~48sOM>~v$nh;`{JOz)`9ACId8X@9P)+6t%Jw%?`o@afOpvIb zdr2}R%t_pRw*1>5!T<(``A%1}=iSrOEFVdVz+>p(-C?!B0USXr2xc?RlMDIOrnyi1 zvXlN`AF#oAyS(XUjAuq3;FT_x9c)4X)M+d-NM1AAGuyu`w>;O{)%Q6Y_%>TbcG|cV zY~?_1W;SGy(r;MvVyL0$*@C}UU2zS9AD+=N8yfN(p$Y$FwJifafJFu_1Yne92${FE z$9PB}1d=Cg=-tS}+Ojs3zFksD{hoJ`o)R+^gB0a~*=w}bmpQ}89HL{+d~?@3?^U`w z!OF*`XhH?X4Dr|UxZx+IxEO&C265^r4RQD}9xwQ|GE}(fkO+^`WEfz4!w>=&!mdw0 z9~L}GaU|8;?>s1yp8K{MGDh}0JL$267I&Nhdlol_X=MW?FgYFm?ugD(-|DU(bdc&J0N#c25GBbFAjHT;R6u1NpYYv78!QH!)t}PpTUcMr3qnbCcz> z*7WoeyY<+NG4Ji*RkS(jaYiPz+~W@Fb5W*l$M?F0^Sf+VC+!CjL57jjjw>y$1Xa$c zd1FcQmOdAfzXtn8(^2n8-oO3*Q@YZD+{C$QFyS#b$q8fCO2>;^O!P`9=eqN1f%d?! z*uXX|d;zK0E3U|HY_M*@hJDe*8ml@wc#e7ObhouCO5oyzg`sTTrp;;mi98?S zE|gWPOBk?f`AtTpf-`)x>U8()=@DAe-rmmR1{;5yA3pknCwvSq`9`&MSHtd(&E5c? zBM;nB#;D|1;k3x$*ndE;;yOqv_wufpVrlOF>6&x7ToRt6M!|%bmF^9~SnsRPOqt7%Yfy6e$?-Zd2GjE1-w)O-;i*>|)+r%7p$1Z%xU|~gG~SLt_WIg? zIOGeA9o)8Se;xP)e2ji!4sydS!xReb90*Yj$w7&_6%}{-X|cyzm`)us-?Kz8P<`7W zJiW~jzaLHoLUR}5woSi?H@Z0e2pxi~S@H>8$tChw}G9OR{TzPzz~aLmKNGX`sY) z5pREy{}+1k^2FDU*$yG{Ze#fnz9X_k`Hg`3xKY_`p;kJ!RI6j%_KI_#;iT*~uYX0^ zcUUc&&aGFXd6*np866in8Nd58?_A!c2Ye|GLj(ErRQ zrnLy)q{U}r#8uzNNS2+4a2~3Fed0`n`WT@mp@n@5H=3Nb{*f%8jJEnc3I;-U8qDrg z*Isx3rDP`RG*-BRT5F(;UORh69>M;NuS8fDZ!;w}5pKAwEO^lsO05wsgxsMXR8muP z#chFqCKysh(2W}A2|bN67_VoG#q(pwV|oV;A8>Sdx0r^e=U?o4{esF z6!nXIunC3^>4RnM?LWJslFEH+^LxL=hq{d2O~Q@WEk42wFR`4__+9t{++0tza0g4Q zE3Sku$BOU{H4lJA+7AVBuZVuNX_Oz}K-v!a3Jw24kfKOE^UCX>2f{Dgg~ER7*Gt6g zl8O2_zaUQaw1Gt7s%s!jwLeq7X{3c$XH09y?7J;MU-R zxdQAg>Q`r2&aY9M3y(=^ywYJWD>)kjX12_-wj@Sm3r6+Mjlw?tV&y0T?c_nn&+P0D4C^ON26 z%Dy50f?>0ejYHpKpV<CSsWq;mG~vDne|?-L z$}Ci@)|NbD4EowLWNED{=bObLpz{UJJ3>qD$TR1Hqod8uWp}$u+x`kkmzmS&dJo{E zHx$=7fTG9n1j%{V!qTI_L)&qz$Vfub>wIf!s}_+$xVD%(+*I;AgMR*;RZ_C{zLGUw zC@v}U$%IwHtVv|yuL;Yin|$Pb{ji(@!@r*9^kppXh_^5rqa^do?t#bB$RkOZrmA_F z2LH`d-t;`-q6Ap%HCv-gxrq%1wJoFD9^){dEvA{r<5%o0qo)NeTJHk)3eKECbJ!$F z>O;rSlA0@xTH3V%nD(DdEXekGu?PX}TZ5D;ELVz!M+j%>Lm%KlgioW>x`P0w;aSo59mxCN9L@QZ}B3y8b@O$Gk}M{aJ2CNH%fi9S0dYXwzGp zba1(vG2_osfNP5;8j5Zro`$Z-5fIEs&kuI6s>fUkkG!CttN}vmR|FV(P??S4ovO|O z*NbbxU=LYBaS%?VMbiuC=u3-UF8xC&oj%NM+QP{fOvtZ+Vh=T+)NX>3_BLdf9# z%(eI&B`PpvG@0KAB#iGGKl)jEDplgj-xV$#=bWA>b)yGZlQf((2z4VM591t{IFJ6q zv{U{zF(i(lcO>iX8nY*sWBOhDkqxGzE%$DDun>l#OyiR_;zuOep*wex_mE;p8#CeS zZkn$K&nryOA<*pSk?6{#>t?Pv=X-~Zhfp!iq1VlJ+o=!v)*b2kQq*T zg`{ijZhoTL|BWA4EQ4An7S68xTX41x9tBd+4rx@d>d!(Mug$${)ZrH9J zy#>ku4+&&mY*8@mvBSS^QsvCa!p_`|Sra{me-1Hr-hTEM)=Y3p1)-nU31J8^VCc>B zP<0ni`JkqgCN?v?it~(ZzVRl9wkC_OMo%GDrYxyPPbA_j600*A-|41*|1j5qdK*gi zgIpB)sOJfcbkohh_~JwhRzrN{zlpQuGuhvI@kHK{fqdvwj;Y-C0FK#V5bhyXJy>Tc zJo)hJesrvtt6pCjVsG$!N0Kc5EedVWI{uPh;BGtqaz2+AbNMqY%L)C8V<^1|wFom< zv?;y!g>SHpW}uU9Vpr7SN6q_z{mYXC2DjkPXSws82aj6_W=I%6DOZzcQQaEsLl{Z4 z!IBiq7@A9~gTa9VKAbt#q2NmDcyZFGXiRwB#9I02gRg{7@p5GaDxQYk*Xi&1Ze(t> z)_v+5^^#5b=Ku?XSwCqjtrmlZOMy>5eldE$7k@BkI zvF)*xfLI!~KZyjSuH4s`?Ep9KgP!H+fS3uZSe6OblJ%Kcn)(~2UQTr)wKy*12VWd3 zPdCOLjizY{%vd@3;%R(Yv>x@na=%H=>^A~x4-)*P7l;`76fz(vy+GHK%5`i7uWiYF zKBDbDv7?RSLjR{D@Pvj(vxIB%&~UxiAq)CYi}^wrUJ=Q^48{`3V47GRrDh0-Ks%rp zNSm@#Iuss?5`q(7U2W*8%C~42{7{3_O~0p9*%uakiv~Uz60l$XBg&eb6D-4 zahM|kk){dZ0!{nkn>hcdcW{Plva zZxTU>4&BTl^ppCS^}LXTn#RY#eBLhBqs{MW>vfg0h$F`o2y$dizgB4I^e$%7h@-p} zz}k~tKGK-366Jf)u3+b-tLGBM;5dL-MDlbz!b_B=rTmRZ6eFf?vk+#P8a(=`i>K=M zjJkO*h+t=$wk-)A=Jax$zI191iofR9C_}Ckg=2*Zqh(0hB6qs?+k@d3*!2C9;9DA8 zCJS#@9Gxf;ceQ7sNhz4pRD9~$`3s3!dXJWR+V26DP)x{4@np|DIGPqRtMm0zcAlSq zQKn>A(xSuGBz4Qcc|&D^EHDw{1AT*CEUE{nsnr@2`Xw~O@I9Y<%iqL<$mu&Wrzw;$h}dm93&ZdvOf>);2cY~zZm zPbXh&V%`sgm`~MfZ{t#n4ddbe(*Sbs>}hfO;|#2hl{x+Cc=JADjx;!f4^xYan|$7s zokKM9e*p5WXpk$Nh$0@w+Z(`Wo0fQN!N@l3%LpZypgWZje(_I$O(TP2`jBlPsgCjZX`8}CLhku&w&xQnS`2XVG zkMMnct@7`*ebQuHe(hd`jNG_dc=1%FM6S{_9+3ZD%rvH zp8png=^a|ML35y&7?G#lR|W`n#B;B*D?rsX3rRU`3y?&E>;SLMu;v{2Ul>qtHP$`hq&~I%}D+<^9EO8lQea_KZYHP87FPx zt$8}UJob$T&mO`Y(FrD&_rj=+%KQC62VyTBIx&pN2cP@GoyEr-Ydl(9_OO@R6sKvf z{0p~DPW~Gd3xnuU-9PQVgtWY1&31H5z0sC6j{t3%#Z8rJ(^1a=(+_!?S=d^mf&b@a zdf)(IYVmbV_F*QLC)2Yj+EAbXFxolcb{Zibmkw(;z_b;axCHl_>)@|defeK(o!-0! zQkc8Va~=5GyTm{0`pTR$hx6iuIXS?a+O02R$+93I}uUaGYJ(Lp=ZefV-jq+ zUtY4(+g%tjO?>_Jg_$UN3XrVlx+)j~$}gD)DNE}#7=<`ipzA-j$31V?JiV1x2iUp) zTU@iwSimbx`P%QPCb_czk*ixD~VmO+r`Cez$8_Lz5F1iE;Ez~hK8j5B)ql=`}{A_B9Xm5clh&3oZtnjhbgHs8M! zLc02{127l#UH?nJyxI`S}S zf}#@zFxT{M-Lq%jRwtw`2?aWrtKD; z$u5cc?wH^?g~74u;G8f2iUH~T_}wbX?DFbaO@uleu|{8n;}0nZ3-?47?<_u_M|Zgf z-=Q5-LrV)Mu=kWYoQwp}#f&w7P4c~7H_jAA+y44C_S*jJTB3$~&dMio zs;Vm^vo@Z<)s7C}99O0xs~6~3d^sn_*a{;h+U*3$nLBhb)rj(Dyeq`+z%Ax3P0%MS zX&pBHZYh_5iVU_-ihj-AFuE0Vfa+JtBs=X3C;JN`MjOMvZx#=olY0f_Ze;HdpV2gh ze@_66wM9S!B4}1kSMtsIwg*m2^US~DsIoEwm{w!w#y}inn5pDnE5-Jmsx6WEL%3^W z&fiN~Whfi`UDn2(+w4rMyzCqPWsW8t#x^~*%mRu6f9;TBkR{hw;rGbXUhSjkTY!vM zUQLK&&`5;`xipLH85L}qL4Fx&LSH{Sxwt5Ik2C4@P>jH{|A$OrXXUdSF3_*Oaz{jF zg6Yr858-y`R`O^ z>83w7mXQI11W+rqzw7*}-E!n{$E$k;Yishrky{YJX@6O(YFIeM$y=LHg1?>Ac{up< zKiDWn(DktlcPcL%cPiQso=o@>FOiie&U948E|%Mx0?1zg2#&XUrmfgezHG;3wD7H* zCrlYHv4XzTmIQOM^ohh{2p|oe0*VE;C$WT$jIBe#wrxqzTWzPUE+oDPF`SGITO1PS znT>T=v#b3E;9&Ni(ktSd6c4xqFMF05#=lMloujMneLrY$y2`D5{1h+o4SnfjX`S1p zANz6Xu`;W?#%Q2;VxY(SNrELZrChn2q#)6jR#KX7Ah#$aUfZQ=>Jg#*~9)jyP#*JRL5QIde6%TXOd*wSpiBC z|7`h`fPfKhT*h0&Tf_0(XS`n^KCfbP;pE{>D*;eh7*Q}?{p*SI8dGatFw1>-N`H2K zC5d3n@Kk>A!*CN%?U%>T{^cp3EwraZhPc8_>Hg?@_dM%-_+LXME74NGaQ^ZG&h+2M zKG0S#-0`NZ<3vkg@ss2T($fFVtJ2Jk1cconwn zeN5@Z%J9@iT*EJWcYymz68%S=f-NA-V9EuCwz%eOD7-ShGE&q7VQqk@t8x2fLj6_m z3Jo4E-DsPvqbKM-8W#xh1+D$7vAqw|(~c@Pc^h&23L*z)gpfuOBS~5U9ZfJ~31t&j zVOJz|gddB5x_&O@S9eO{K*P+(4J`XgcAQM%?l+cYh+>D0vpL_tJTCsja;lT-%#Mn^ z5yf6`J4y1xIjF*;6#1vg==T2xFh(Ka1%yxGdJ8h*qhD$~KqhAJ$e?Re8O5G4_&apnB=1 zRpN!&xfHGxxe(Ukqek_B!T^hm!}%5rY78M{xB(++$EH16z2-1oM$zGUH>EI(6p@fO ziVWG({ITST;&i>})Zi!O28pi?qJUuaJa{wGZP+>Zs6RAn$>JMk5&XaU>sW+k-D5T?=3kXxwE)nib zRrJv8JtXyHhNyMDa8-mE!+fu~JUtTh<-Er8&Ec$MlmYf`QoB#_I2Sh`$FaD?;sAwRT z=L25(qE?DqFg*15M*E!8u>>{V0Rsj9PqWr6@Q^um)NNe**=> z*>FAhkRnLe>Y4dTgnD945^TKC(3aZGB-c6EJniP)3wl0SKc2;t5eYhXI`5zV+IJ=7 zdlccim;Km}hE^k{+G#-IYX&W_}_%3YxPbG0qUY&-Mt-ndK@vfCv zUN{xombU1ph}|}t!_}+Dool26>{jZr|6*#OyDvu;XJGh3MZDvn#t`=cEQO#cH9)_q z=V~N1!MtE9{fv}DU_)8yjF;GvPK}0JyQl)PIgo7b%?gM+j_n{a9=aEvRr+!J+w&uJ zAlnHCnh{I`KKZjhW#Nr@yC*v^kMwX7UhXXg<)KS4p6NT{!zz#OKdA#f~POL(ua9-XlLoF3WAFLn}e(OHZ*U$I((q2X#ewz=j{ApYLXSNewb)9MhC2db(~jZe4E{x8SVM>2or;d{G{xJ zo@|p5f%&&A%)eDf%4aZo+F$q!YZO`Jg`#_7Xi|T7>6c#Qe}7P}@|qO$6Y07Xtjt8L z-A+O{e>%GJ7}Pw*11OzFiY@p=SlYNdpsQ#D8x#!Ta|{v$M_|hIb|~e3atq?I^D#l& zz4ylJ@_$CF# zt0)5^8$0CjPdh=#y+(KFCuViPz$3OOtcsX%>x9}AGlRWq%exr@9>wTaQBC|ih2UYORWg4Ks;0Knr{D_-(bg3rO%IL%nvL;*GQ zr}qbmX+e!Vd;BUJ_ucsW-;Q^Dd-!$n2cBbO3fE$TR&y?gPj}n!iCa>6Ji-X7^NGUv zr(B-^r(=Z~y==-@g^MZK@zBf7Ja;@*{>0v*{UL#KFTVyjr1o0K1TAHbuMX&k_F5xs*c~kbp z-V5d0r(pX+d-(MC5#mAO-|%u3^&Iu~nA8z#(an zTMQQX*}96|y^MiWZP~-4zpjCPUVYIw#||8f83c0qkRPtbgmV$f6z9d#opHy6>sOFk zSJu_AkafeonWl3`iHkcrL{$?;&pYksDr(i5#>gz z4p^`5ifc{CI1blEJ@I96>WFc$xtl7;!a0{RkSi=d@CS-&i1tAMOUHi8h{& z*3bOTB{WQmdhpF~msg+8s7vblnO{RX&stes><@`;>?%sa)$*FhH?fRj@Zs1(xW^X1 z9^s$U!-fRL&?d>DMq#_i_ixT8<}_$rAq%t~%nOtKSutA2v~iEcd3fB)!t^FOhZMrt z>3Xc3;kek?@i-;`f@1oK0z-&E(luK00lM7X?4(N>^8t8@dl`)uqmp`{^FgOYr;r65 zlS4CVWbeVHINS4^w2}Rt!EsCYu0oE)Vz=k+`0DWEizs5bv|pPd+=Zc!Z*P451Zj=* zIcnTy*jvmk9TCRUy(VJJ7^kDm9(0Ur3LfH&_Sx{Cj$VAP0NRbLaoz@V%aixI^0R&5 zdcS}lCP69c_Q8v5OI;|A!DbQ2r>jFCJ26uYs0tVB@jZ!3*dJ*Awjk~cd)hF2P?P2R z6Y_gMEetv}e`cxbyHU=EQ|F@q;NbroiogS)S z8IAdV(r)p&DE80R>R_DTLwB8{IWuAHFEbvnGA1_De$n?ou8 z7}e6+LsL#tCqESM7aqA7X`g$KGZQS`*H4SFjZ{J3^yvJ=!!YfVYBWNhoMP5V6k$I; z+w|46rTL6ee!)5>J#pF(*KYpqL=Ek(({SF{@cYkVhF%0si{HlSspD8WW$#IUy^rOC zA`vBxXXHy#{u7Fqbjs&6PxHfc_89p=9Hx~ry}vlZ)WG^@>QO{CgP{+&8KlGpt%;9y zc-hH_B`LEB|M(;8w)DhbU-c)rw^QQvz!?Xr!dwVy)TX&_maBvM3vq~2?po<|lOTuT z!dV1Q`ZH55)69bm%@>oDUw4K{?rNdEZ8Ysnvh}Fy3}T%QK;ihun4-EkQ$WxE^Z7+ z`ckBxB|YaK+!zB$TXq2!JTwv+;*zi$m>r#pKI2I@5kAM*Y-wTPq=tVUcL<{)6!K8 z#@<@GmSb8Arx|0t@#*-lX{;vjX+xPj_}x2pnQvnW2#>usj>N7TzH7*>e$_)4YMbiWB|Ae(_ZC{Bq5Y z?{LS;12DU6T@vENONe)@!&Lr1TngLFO-~EE_c2phnt0ASKXK2*3rzy zZIT+ST7^ee#yipbs3hL;{v+m^7L^g3TTywVXQx+%%z+r=3=D1PAdH94kX%UlG_$B} z_H3GKDp*e_&Wpoy+k;;KKHo_H)a}bhB!koUPKYunWAJUv zL}mkwf$~<;ok5TzY!kKvO_TCNKIhYSaAmDA_Wekxbp4W9W4tHS7@6M9o+amjoOqg- z7~8(wb;4m%?RB>I;$tTF-(mPM=rCH#?A1l z1=rGV%Cw=DoMyow?9+RLBfV0ScGidRu1vQzCB2^rUH8*_Broy)eEcV{`7w;6eriOG zd>VmZQBiKjmS1OiFZ}(QHq>N<{Q*-aLOi8^OPHhuPgB!5Z$Nm-x`N|KUKRElms1xG*FxkfB zWG0qrLJfx`{XOo~qhiL~|;3`KuRP4WKR!9E-Zpj)ERqJDdN_V_h1 zs^a?I7dkfuhz&p|iKSLxjU*Av18#S8BpJl4+LZk5vew18Bu2dz`Z$1b+j^IR>{mI9@{f;(DF=;@jQ|Pw4+plV3{nM&)A-hMcd0+` z8t;^(j#`RtjflrIk?gtf+NV?2v9e$|FHB3*{@&&8nbx`!@<36qa&c>GSoD+Ywl?GF zjH2+8%IAIq=M=GOtC?Mo&lg8GBbCD|t@3dsjwDO2bQ>@FBJ4>ZR@|PV z>HNwY>JIHeUp@F9TePKe5lbXRxl8p!*S-x?TGn@{x;@!+p7iUjKqeW>o7OxNI{uTfAyx zw%_Er@ysPGv1U*p?toQhM@;Ayj$sH5IFNUcxo(hVhz1jPsPYw{mmnHah*y1A(;fmj0 zQ8N?9O~b#)qwl+7LCK=hBTREC$ExkH5AM_c3X4|eahQ6%U_>MbeoUt{E5H3pSi0YF zj5z5R(6-sIpX?a?eRrF>bw`lV0eKhPV7!L0JlW0S8tU`Rl-NnMtfS42dd|ySTRz}L z4!Sx3=-CD&(4f%NPW-#y(0MDfscqk+8Sq%2g{S|8>;MvHqpKoMaD-$&_vuT5x+|aG zm_e#X?feCna)Z=Dsh)KGsEWXbn}-jZR$X;s_L!su{UL)A;OxME7@);AnKEl?qJx5>inTwLt==|SnW5LUp78$XPx~qJ* zl0E~xWppgqsM^q%LTKy^|N2@G0ZSZbhEiVn;igPSt)^Yk3LUbaHrJ<5&xNIqvai8@ zAB;DNcP(lKIJG%^QmJ1=7=1FUDA%F6gX|SG7an*cSJn2y8xInHF18sp$vq>q`la4* zTgiU2%?tljU@oct93B%sO^NEpMnQ?LMkt#LLNz8-S+5iNr)5PW$!G{$-gV>qm&J7uxfEuy ztNUS%PIax}47IG8;}{;W+SpY3!}~Nr0o&idyu{2D_CJ>H_J32`K!~PwCFv#ChaBGP zMsNyd8sqt-AZAq~n3%-d!b!DG3Cc_30i=~CTw6itL05jX6fXLAW`o1RuTF55wD2aW zgo6Cc%*<0SO-*n1m|`9FAU2=#TirWy9nlMjA@h19D|w}HS90)hGS?!S&8TbX%p&6j z34Q7T^@O}l%aQf3tBUs-EBe#}I6FjN9qy;RBwWKDoJS}GbSv37VR2zn8n?&FHF|hO zeQMa9qeA9QR`$85^d|h!VYL%D#16tI%2$C)g0F-a1WiDxClTw_dV@Qv9hvat<2@&v z5A8rXgJMK1WuF`#vYsm7rTq8&vZI3)9GM zgj6lc`DWV;FFY05N$U;H!;A?@R8)Y=Ky;Ix8i!`=Fg;L8QjqUoE(2NWI%s~)gHJ#- z@Kf7Fi_9y<7*hnJakAl^cRc;+A(BF0wz$5G!y{jN>!e=#Se3gul~#+#2MdN}tM59@ z16?MK1a+UqxBd`b`Qa7i7X~U|jV-q~6gP3jPkLrGAG*Nu0E+z*lD{l7O)&8EG(yec ziBBeIjEk#`eTV5h6+T zAu60dYp;AH`$m*TkdY2sMu|9D%rgZ9UOB{+B;*srI5X_rDY6x zH)IsVEl|jC`l;MHwxfTmSJ#1rP;5XZVXV*_Zi2!za%&Ca+p5k>Li+&Hc4sJ^NJ)1F z&;D1T50VRCzcGZa!h(!#sXw0|2J_U$R5LPi$%f=6oFssTFZPH84 zIAvhhB{~HHohf?)lp&axPr}-|HlE2hrmos-o^Dc~NV_qkT9hV=>AUcKmx{mKB$0GF^Rqup zR}iRN;oOw3Ok?h~Pg*3d+zYUp4y$yZ7$IF+b6@;!O0>mjKUH^uS{wj2@}e0|S;(?D zeoVBHH#PZk^F9*#XuTlZh`;)|3b(7&Hlc zT2g&|-rw#|rw9crAnwI?e^6gFAu=?SpG!=(*#i3=F$7eGZIP#^ z3t>yflB-i4)<$`L9gETGSO-iD$I>gUrw@7k_q{UJ@FE$~6{T3>#e{j;vWvqR{9B)M z5%ZPMmoGotF`g7KW~;8uba!g@ih1-P$BEd~vvhLX`O`)ommwmGJP_ zMmVT_@pXlWvK4=sM}%Qh^)~A&$1Y5kJz3Fo+IY@9+hq2TPt#t>yr-f+BL#G)6$TwO zHIDGZY}qCzwd*xJdJ{=>Au%*FgeVM$6vzZ*VCCGa+Lbk5j+Y(^Cp&`gy~;h{fEVU* z>{EP(@IW#=WK&FmrmW()RLxYISC6$nFF$Iu)V6t3h(_$~LGo*Iq79FOEV2eHZK*y` zE0GdjV|c&i-){iIgeF62K;1lIm#x1TwjMqhYJ~ljTAp~GW3tIBmymVi;nJwuD)5g2 zwsusa{95Z1hh|YTMT8UleIv2a24!>5x#x@k|1PPyLN`L)Jr+05)WJ&osP zlQ^Axz@nqTI`Y@>hUVRLqjai@1uU%t`a!R7wN9 z^o5E@t6Sfzx^s84GsW#;CisOA{`|!)vj;kR#2j$sG&Rf>)zo;Sz|Na?^qH41c6UdrG}0nrMqKjq!f^D1*91|h8!7UC_y@h?(V#UzkBcV%-`>G-t*4c zXYaMvUOUV`nsiazYu~=%2xr;h3J`HXNFeq!z^(-3`TUmZq~uuzMr2Dt?d+x8X7l$9 zjp~&D|Kw=gFh^iS5wGe7#NVIyPU#C_Xo(07sd5w#QBQ1Gm#|pHL;dcsY6siZp{@pF zyPQc=*xL?;Aflc2=(!i7kAf!DzWpKFx_Dv_IVG%b3b!&(6KfWQu;^*7rSwpI9iO@( zuPM0EV#(z+_=~@8vnt^O^8;+4*N>Q*o-miAbWsz`(;Ib{f&}5RIL#L z&)4|PyKL~s%h}*JNQE{ouGfq5iwG|5tlPhKuD;jCysG~C7_=iBaDbk;=%nS>%>6Ac_y~0&ryLRr_^~7TR1?} zv0$ayIOXnuCc-hjt`Bl(_o6mA?4KZA0E=S-`$2Hxt|;cy#U+-_Jw0D*ot5zJ2apod zH20>Tq9s&{-MxC@&ugvtpUw;T_YW4)^=OhD5@zqqKcOsF9`7uA*X^lGQ>tKZJLSxG z%c8o4ZJ2g^H2Nro`;Er3+=9(&1gG~ZeCm;0@VJ^-bq^G&qM1>dwR>-Z;6A1Z zcs;LqT+N{|%-U?e%F;dB@NhTiSw3u^+yTth?~Y9XwcE8t(>@z%@+3Uu7w(*!N8rJ_1)qg0e=Nq7_6_?Q5rMtO*O$Ido81jxo-2oiRQ5%-zu}TF;jZ= zWFz%9n`msra8WHKAd-ANf=e86J&{*a>ogXN)~`9%`$it%+4_vLY{C8nF;N1;Eqf!; zg<%;nsMskLna2|+&FFJP#ovWMc?f)#Gbvo*RdWO-!V(HgW{A)V?Hn!}#l&~6b z4#wahhe|4L6JP4n`~@{dRHqOj^AeAo9APr+lVlyzv;Ng+p#ZG&vyrFFJ3_3ddW`@P zW{ClV04fe_+T$%lOlI-uGw-fp;U8C6Ot$`yHH-@6c(aS0c{G*ULv9D@!!Atm6D-!of zJg3th#wK^^zyHsFlZ@%M+%f+(O%@U?o;GoeRjoEhZ3M-F3oGR5UE)z)wiI1M*bMQR z8tuJYIU|;_ouOUwF;Q%xKMI9qrga&^93uzXOb{hcjnqUI8+Z>l4Xjxa|9dk3BcMa& zs@*s)!GsWyAdRS@@OmNetird`BT7vDmn6Y1g|&yF$brlyNv{6GtT=!Zpl?`nTd?T= zgQS?+Sf%ZV-7f31}9fS-hEa z&8aQ>KdMz;E?Y=NX|f0i60Ka4xfnPu>lx*G+dm5!$A9)MR^j-WEe6tqU=j7SiJ*+o zlP94*XiXH^>7)$o|5Tw&f^2t!Ps^k(?pKvb!>W~E=3E+V@ID7>-yDwhSWW)AgzH`6 zJAHTuHFT!pC|!L=OX96S`ouOWgE=7zgve)$=%Vx*e;%@%Lg?8a{4bb&qK@vYqYSG! zo9MmC#87FrJ8`M13ATTq^4T|!QbwAoDT45pe~~@^Q(s<38}fi<4P5a`f5C6+?#+!6 zcdB*1I?Ympl0EC=g~SvldX|TRA2Bsjz9;>T&PlTPBp;QwY2M`;&-vU|Kl+Mohv7x$ zw5bh7fX4KNZ>E<%t4WW{ShEH9d_Gxto%t0)xcs@M_sH&Fl&Dlp^L*aG@p!2y?_)$y z(u~@_H1>;GxvTTL+*Xl0*|tQBPv(Ae2^%O2fx>ycy5ZtD&tvo_7FulXOLH*T+D~?= z7%O@zDFQFzm^U9L?uUBJa4mSFUe&zGdfv!b=82^1;@z^ZF)7=9?6Rg{>ZU?TnmV{) z!GjXah@V6n;WZ?-*|ELb(ItA8=mu2uI!nWn$ntW8B<(ZLiKz?oyS??hud(zk;{>!3 zT^LKl-_|uKJLLI({5zs#W;Ibc0D;mEs(#;EKu5OKDqphDK>M(7h4&$-o3yR^U?{`^ zdmlH-^fjh}1Bf3d^xie=!s2QV_72PKFP!8U>4ER~F8OL_wko3Rp9-AGg6oh)!h7Tp z)xOJYk@nYYv^(nRF?@1p=w0t@UT}!$9=;xVGXm`lbFEj=ZRqZNdK@Q4fr^j!;Iyxl zi!&vz613ba{a78%GOn(a42XqVHkOb_%aomDD2fxZ=d<=Z2g?l0B<{>ifvQ#dOCX9b z&Ah3E46d7=DgZ@TnEES32`F^x*5_~LQ$j3;m$v8U1{r(TXCnp|ZnK3o*LGVMmx6ao zl2H+h(14P9=jCvw*cqR$<`Wxe`nCk_X35oy<7Z~GE(7VvZ_%d+jc~_L6Jcbpdf4t+ zt3-2ST8Qa#&T6Uy)qPfC?+9*r#R?@&GkEt3Ea#;{Pl76eQpF~*Wo`bKLI%{zSOel@4VEW;va_x5J(3dy0hBh)S#$`a zwipmkiM_#rGLFfIC*tTu!t;Mx`0?arwtI|MIP{U}vW$19o;U$FjuGe&COyfADXBaA z)K|OJaxe_L(#CIn_J8^u^UtRugX-(Iqe3lCLl0soiRtms-)iCgtMmjWc(j8$>4w6_?N$2 z%$yj*17D!Lo8_Jp@srZG)sgkWm!-^^)OiC?&w{$I5Z#-y2`<62?-$BzDVMJQX?6u{haIAkIM-R!F~b ziy*14{Hx#mT~Lov6>6e?{ZNA(0tMvXKUR$X(&8Vo>@FFNaJ_(ZXH^xqY*{S}1Y>#~x8t;n^L z-kqZW#UJF9kk4H`U$n$|GX>87C5hD1kdvbiDqnnsi+{xvLoRIJ_oDv1UPDej*jRhD>(_(}ZgQ205} zeuPV6rYX?B^h-=Ah4^U#gdD>+!;FJEO(v4%>BLJ$%!l{QeE{1QrgkOdhb}F=mYyY} zG~IvqfT3v<^KIqTvSIb7L` z{l1AIFvN70M#RKdJB7HCs!_Un2%n;((%kMJtZa5nIq4x0 z&j&f~`VL>$Pzh%Sqy7mgS(GHx=W`>+MYJmcEteDM=1Yu*u>2xWgdQ4?um)!a9e&v+sm_q zS)^TRp)X4(Np(NY&fM-9t~pmtpM9FeN}#C@!GF;FERK3NyiLRr9?d`(d+`mnQv}LE zBl}$bB1O}yJOaJJ=2G6CS^}KP$W9G)vOP9Q|A5&Jh>MThCW-J-|Am$+QMM|He-g5x zER2|6kM*3q51c07d}=*2XE{C;ZGLv0C-1Nf=z7&Eeg&#Bq{EBmn0sa;)!ed?_4w6` z$p4>9rT)rQZy$LV?n&4$-fmkLY_zP99!;}E7UDuc*a+#R#?LG1);DRd*Qa__zpZt3 zl9rsy(CnPY3AAcFI>R9*iKAI4lAhu=R5{oTHv=02RiI=x_pUGKSP=MsYI$wy+flxk zp^T-10Bv+m+bR?S{=Pn&c@cwvUaVH%aJP1+nV!B!cUqIEq;OsWU3m| z#TGq9rgIHkxNhrm-qSmzP(5e4(ZRcf-*zDW#OHih$EyRqG=EXO_BMaI8DKe#GYc)_ zwA~w`Y|wLCw$-17>n~n3Hct&Um)zc{ZILSOL02c-XpDp0XjVi^eQ%qX)7%}VADSSe zzC?~St8L99*~C|yKZF#AavrK79vjCXxU;s3)OsqqvD;Sl+xAE2lEIk&&uooqGCC$Q z>-!AS=hkBHG-3=#XX$r)av%j1X1o7y>qxcA!2$Kki5QB>Fch;ksOJonq&g@#QuU|k zn3ApfHn~)iEPBRRo=p^Y)&ZGLEWHkBxPMvXf#R~TjpWAc{l@HLS_m3f^1lw=#QH(F z;y18HDxc2vI`gajZ!wMc+WHR#`cHaTp0kc7{+8vgN6YDOO z?q`$#A^E*M9#TL~WG@#CX*Ds^LReJJu5FF*h2IxRB}^87sd2P^ra#aqCbUfrvj5l! zzHK%LbWA0Ri4`Db4}MYI_Ea^|jdUaj@CRRq(xh7`c^3V$XmQb%Ia}VI?R5t)D(~T` zFTP5wR{31>bO!L}eRhs{r~9m4{gt9(Ys4#Hx}>FY;)c%oJgQKbzSirkbcol9e#PlI}@~G9|4vOQYxr48rw- z49osaZ}1$XHi0#nO!lNO@aQJLH{=kRP_}Av%@s98PU5_+GXtVL-sID{-n^bVzD}`&ZF8e2M2xU>8;iC)Wu}68WR3-0ovkhlRcIFvEgrJm(?fK*0UBgeQOpW6{8(*FWZqloL)Opn; zbj012XJ$ybpYFPxEGteYX7UzEOBnis4sIGdSh-_=8s5Np`@$vhv}XTMF*fL_YIK>v z?Bd0d8Px;PE)fTu9i~72a+Lj+HcXNIY8UN87|%yZHE$j(;%Gu#2-SW_wTe>o#>$T- zKf-}c+#jE-v!h_T;?@$RkW9wLv}Q6dQK*BfwN`!wuh$KO$A7ZfVNmk~dW7dVzSFdI8_eQIhSoQ1fps%OMEdaq=+@dqgFuMhMOEGxWWxI=$Q>%|l_4rx%RL+dG#9s!}C`O8P{GMAhQC%vI^>!VtO32TX z+j_I}li{F?3)hR!g$QVYoW!0>uNYQs2GRH6!liQf9#x8LFkycpnYUorty6#r;H18R zk(A}BZF`Kz5y#B(D0YNU1!Y_{X}A!46yBu~>hQQlR7k50?d+ zA7nPnQ2xx`d2vZETpjoGL6?okQvi(?V%JqgF%@LdDp&@urioyzD#H$NTJkl}Ua1oM z*zalV4jTXZg1Mnbt)lLPo>by2<0VQnj{5DKN}%nxI5}V}z@y7?oD955o)Yb28ShN8 zJ~nM)Q29x7ekqSqp{of@vG_!aul7yY*{k6zJovAVyDrW-#ZcO$Ih=a(K*nR{Lg8`ZS6hCzX< zK{>>xHxAhU1G+F@O^a@T?000wCPl@QB)AbjS>&@n$HHGPPRwR}FtClOBh~};*LVX# z;q*7J1GC-HGs;QDnSMPZgj(Roq>Tj+@ydDT(k45=M+i49^JuGR9o`+J!zVgoWeKbu z`z4yuX{n^yZH<&u#nUSZIcloNfZ?WTs?`I69OB1fC*BPj3Pata?S2JuzRYLObJ2;y zCMi4Yv4iDZ;&%22$C^~zWgiQF>lt7Sy(4#gZES`JvjDB}&nNguD+l0Tlclp)hk>&# z4k^Di!goJ<74xFc)@w1{P`bjCBm%o>r3o(~Kl7h&7aLlhrB5cD@p9|envdZ3Wa>17_5 zChyGjyQ~a|{g=L*i@N&_MWN82TsCAguK#gWy8{@ z0h%K5IgA0u7{)VH#x3u)_p7oiFWCQ!FQRWVF(&rYNnE2K|JquJ)gsC(md@OQBxl_U zG8G17-OlaMH@FYZw2o7Xz`M_^-UYNeTTOfy@YJ((-276&ZbYufGg9`d+f}jV*KdEH zEelWn9;$_Cak6oNj9z3_$&(?-;T!U!5qbg%Q&X803Ag`o1Ca!DDXom(fO7h2Chq5Y22)BcLQ z36fA9rVY9z16mRyoOCJpIrxbXD*SDmf|>77zUvRzrdV2<-#O`&=Nx_}x^E@dy20`? zAVw0R$&x&^eX-w13c7_k^f$sj>Dykq*<1P4-~Fnlt)R=bs@Iqs>*D78uxqvvhver7 zqxxd-+KrS$Vq0NCtM1|Ia8b7M!(?Q~^$q9gHd)MQrN$#-o=%2c@k$^EaqBqR{p#cq zz5u73Tnv9Sw99>dJ)eeUzE-#Ne8nlWjt={jsWV$m*CR=(_pPnCM5S{FcIR zZpTLwUWtE88(VO1s|@xR7H6eDkLp%>2-UH}4^p4-sF$1#FfX}FY}iHL1+Q`7I8{4s z#I2L&_EN3urZfnZZ_Ha?_K01AltpMn-aOAg0HV42z%ppw{yY26{;3%fk}~O=n)`1u z%`UI9a{$ZW^#f(oL*&;~U+z68(K%pTl@sO&zO|+8gUKB`;DEE&!zXY{#J!=(Dr@*+JqBYIDn8YROel(zo5<@ z+6~$b5Ya@{Yug8J`oZ>_r$=wW#c=(+DNmwqem1G#D1-x`&K-K2El6PdkF5EYoCH^@ z;Q2~xt(V*?*JrNF;+~2I2}Pw78oATWtUO@!Io`?92q`=L)gd)Yt-nK%)CDW($HQqI4ZHy0Pv9vO!Y< zvANn+!;k#EGS&7z`jOVH6{S?X*KxUJ>55VxnX?m0!Lmr{p5u3pj00=ti_gh3E6g)Q zIN!K>bCR8WB5+*(gh^&mXKq@Fw4)O{)iGp8{AX&vjJsmKSdWYtvM-|*)ADOx9bXL0 z`(s0w486YLZYH3*N`vg~3tMb3pIkrfL*Mr;Go}?gjd^9UqA!x%*aDU;f3-++Z6b;{ z75!PWwr&Kl+1a<09L&#udgNTbLwShoK}Vp=h59MiSRlJ9#&m_K|j-)1Lp1&d;@I}RZC(hL1E_%dUQ;w*Nr z51PZ23H3Y+6-pqy{H;IoROXs{KE^VM1!SHvJ)B{B6zD{O$UNO~h(|bitpy zH)+GH5pe=iIRKq#dRT7`)q70gd6N_BV9FQo$_2AA-Ad-NB2Ry!I&GJikurX6 zm8i9_`_KIQ94kVA;N)J7wk2(aBh!RV&8fBQ33*Z7;8Wk*ls~qm6h~xo~KYGKQ4K=zkLl&rI0;5mbBzk!;uWYD)3x!Zb6~scx8! zlG;Bqc;@^4_Sm<~@x?hNGdIh1t+_hBFGh+eh3CHY0``IckNC%a#B9Gj=RYd=4@%GA zPm!c9h4I=WQ5W=~JX#D!u{7W)uRiIL049xD`_HzoIId+PjldF-PPIau0e_`po3ubj zd0W{%gz#^IKas&`=8flv1;%nxjsP=Pwv}ZRsh3rSxS|Ufbe<5SQ+|)$RDUjnl~Ckx z_W|b^?1!&oX#w%xr&^~gqgWdi@ib!ZUVvH_T{kc_>cjDV4VbSThwZFd= z;!a_y*BB8WrroCSJmB#;T<+c`s4ptS19MM>cyrd|Yd!lP0*QaO|FTE#q{pe$+wZYI zwqI=g$DxCV8cp0u>i_^eA5+9p3Y+Y(Gt-v_Nb{xji%l^IW8{TYK?vJexz=(_lhd+HSO2NSVAQLd}ceaS3NM1#sdcSaElCp6TtxmBcx6xTNXz6167A# zR`^z`Xo1?(DeG@DsGyjbt}dtyZVNXob~+HW8)Tw0p8ze2fh_9U_RC=eV<)C64fnbjuqOv}7ihFll`T5!Ga!L;Y<9?mtfC>feZgpC zts?G5C&4o8@R~^VxtsE!=H#(4;N(zp44a5vfUD@i{7cc+ri~X}M&{>OSKpbzubpayq( zsvkJCLbXYFTF(`@c`I7VJ#|qm3OiWWWlc#z^pE~CocYDulu=WO=|Lz^9^WMT^1C+- z)G6tv&O-ep^V4S$%c4xi<#7YI)-YW-Ul=nty~j)T(w4!rujCO{V5x8M_v54rLiMW4 zJIdF3ay2qzUxtmFZ2oLicH}xwYXrR|&mUaS{?Kt6jA1~pN{FaVarCZ4`QReJ6#*JX z65sqe7BpfNUNSmxA3G2X;|eh9VW^2dR8gj4QgH+JrAe`oxddWAqz*-}NWk(u_-}&i}#QpZTr*PCTZ5tA1PeHEy)KYfv3$vG7idMabY6 zy)qQj;g6$Uk^^WML{J}DLibY?Do7|)NF<&R#P|u63L2PO-$p|P_k(iBhV{fnpMSbv z!cVJKJjLO*%aO!X%KpP$f#f|*2tyzq;+>U1_|^aq5`#uu5A*;AYsRT(49^YZAxhVC z9SxUA1{8*{b1RCYFK4`R7Z7K%lSo4Ud{IU!Tz@0hbbO8%`k6x`3B@9#65bLDSqHP9 zytR0d4UA?-34c@viChl~vfjHSglSY|#uO9;zFkb7G2AP{pEWx+wr2~LCh1MvISC+8 zb1aLY{di^9x&F|B5e~r zN`&yH7((=$oy@hQ5vmM+Igza_y);lvKw(s}Xgp2^&<5~Hh&?=ApVhDpm^_J6fa>)Jh#_KsmXvpnVZ(Yv&{s}H9LTi`(9Aiqs=4fx9dWfp2&$4c%R z_mLXtU>13(@qz$t{R4?*FjyboREv9<*3dS<>)FHrhcTnGG1Op;Qz>ap%)vI(c*ht1O+xw;>_Gx~H(Y zUyyA#`l*)p1Yo7$&}``^bl3mY%5S@%JI8Eyu`7a3`0P=c8{kKXc4 zHaid0jDufjo$s!Z9cX;sIg&@7T`e}N&RsVN`$n4{%a!T#+Al%Y(+JyjBaKA<&Tef# z>UWM)&2VS3w9qiZWnLUN)FTqR#W>rf3fwilZdWBN>_MeSeRVkPC(;d0)AZ_(1qX5j z#kL}I+*=3bDp&ALICwlP(VLpeMf;t%Y&nZer%Jzw_X<%Ab-Kg>`kH=tE0U6gJ(lf3 zD0)zQh0CNl?;?tb925ITGlyGTiN2Iw`eVjEDWKs9>zvtQ9WOC@fzqg`JNqqjk-EL#RS6?eF*Xs0f)`m(~5 zE1V@k4*uTdZdj$I))gzEc`TC%tcz4LNgw+^Ak`?o(R)FCeY;*?JAy=t!8cPS|B0;b z{C43woTLui{Xr+OgCVptImJ`va~4i$cr(vS4hkTr8m##bLR-7Q)2B~4c z2_7qFeGO3U^Q zUds#7!I3t$Y?LE#7nU9h13>ZF*oFM5D9i0^B(;|ry5063Ipt@bIA}8C6P0%y(icE% zG#pKcbv6sr7lAQX6ejN1!$cELnv>f!Z}+2SW^s5SZ*@p>@j_S;qCDXAEIh!k^F5VE ziDgi?0s%jvN^G|mn-)|?-l^iOODH%#JdtAPV7ylgyFnpky@}OykBgl1Jc45I%R^21 zS|I^gz2EI(V)&~srwBVLkL$w0)A5$usu^x~!jXD?qoJdYWzE^i zg+oW_+O4e)cQFU(xw4OMdlq&NWoymdWMo(SsOZOjG!RwLn%8s3QhZ-h5@G?amtuiK z+YJgQU%*@MQ*`bOCS-pGqXrO3D`DsfP`gCCDuDMWDD2MjqvSqu6BLDGk>n?-qp5TU zKM6b3$G=dxP88s&O<|?_^bE&WdvoCPpeu^+(%hMPZ^zXpm7VC%7p`WpVKkYph|>90 zI>WS%tKT}zdS|M0_$-M{$z4!i!R zj=hX%3{NObFx0T>vz_51aIbo4@%4_mKeNU`?<2!e-ns&~YV;|ylX{+Bt5U85DC!Zb zxH=x3j{~?0Mt|IP^bUm#G$1KQs%#@+WH=Wyr@<6{w#xWQ5p2t|7SesoH2R~cq~-RM z{mNzp=t@1c6bb|Ar0tUQ6V88S&cU_<%JdRmxxk1n3bz;5VWCXcmdL!INTH5jOj9oy z19INn`25si{$A zeq%WkH91IVJ-{Amq2}~6Y+GqjAqNfkqfOTC#hhdPfUUkAa&FePboaU}eiOK` z071-t#W+^l9@htpE+VAKxML1?1qhP|in#kts93C7Q)!>7K^NQuhFsq86|a`hL07o* zj-pF6U*t3Dqe)-${%Hb(Hj-sGtBAts6}Jl(=i;Iw#nD4}2|Vnk#OFVAI_~P^M%_HF z)TbyH&C0aw@;wi#!8sA$NGYtKyCw?Sd>WF-nK?jtODDJU*(c4=WzgOY)5*mcYjPTgP_vsx9}e?*UMQ?XV*Mf)G8FfbAnO zv{nE73G|IIIo18#2Zr`8h=6j%iZwp#lGC$#Y-;``-z8~aAuDI)&y{x>VIvD+8(=XH zUI@IM5!X2|=w22_3jufP>Q14eV1*#`Gt?=5T2Uxi_6iB3L^9cJ+z{W@T)BWhBnYrPKUHT`Vf?Y?vuy; zZznr)v2GKQsg6#nfDND$3c{SPJ}F{(iHAxnKX(?#uew(k^x+~wa{E*SpK}b*3dCu_e+M;JAW7zmx?g{+-|1VnQ*k=& zLOE&F#H(pZ8&Q_2b(4EY>yF`tMIR$dNhgZ8`w?B#X*Tyr)KQ(Ge($fB*-e%1zCX)W zrm?BdwI>5l^$p6Uh(q zGNTI(YMom^>kD+ut1xi?FsuEvfEK^Ov$Y2AwIm4Ys$J~9h^>F#D;8HbOh}XCf4XY= z>R>CyAh0ae1<{3SsO8Gdm;JdCe>T2V_0YO>GVSD5vC$+cc=)j*j1?a_QsA`rbhHK7 zXWDKm7_ckQam%}#Qk%K#Th#D<`Uh5cTY0tcDV9T~aJb`o+}vCzErm?Id{E44Q6Axy%$i!hh6%XQH|Eqvlwv5CAq7Ec-vprR z>c9F>OqZ^iaW4SFe8rw$Cs0RZ7j$h|6*4wNVuec61B(tG^ zTj&R}T(q}eA^&x~-R19)fDU@ndoVat+-Te;joVWFKzb@F;&>CPLB^*+;vBbJ$4B2K zP2xRGH{Y78eM^w5v>>T^(zp}&a+;T60~5-j;LmwF1z&Diwz_zZFYYP8h*n9RLQR`- zbnUG3*S|pQ@#SVdKv|?{sGgEkJxHyVCg_*y)B2nn*19d9@;cQXF)T7#zQEUS{Y3y+ z)V-{%8*t*Nr1tFbEFT{FAjs-g{yq`#}CSFx{BlxZk+eoq}@nmbAa`UYDSTE!%be0j8Y4**!QKAR`Y4+10;{nX6@zI|!a{+(7K)+p#u(`CA z&7B+1(7C*IuVzb!V)Crsh|b7wI2UrnK;EeQvwr6kWrm{zC+ zxTFa5k=-X(qpEPV_;p_{XvBO%MPPNP6!Ru-D$M{FTRg~v-csdu_jI^t?xAbB(DFof ziq@;&iyAF|z+-73`=?d*1hDu9r)yz@d{8gxZw95LRnKp&Zrf_{XOUKd;oyu(^Y)R0 zwgmn{7Muy4D zzNyMesYq!lOq0!#3B!!P?_Z_~ggLSC&{$JkHU90eS6-@j%j)>+zm%0V|Fe~6?(;*f zwb|QN?_gG<7owN<39ivA(+>b5qSrb1x=qu=%I_wLA3x*`;qBNRZNl_gm=*`Yj?Z!L zM}#p0L}LEhmB&w%G|nEqUDzn?0JyXw7nEOP?1em6H2s;kW&ZPU44iCr;pIs-C~WB* zx-vQ`YYa8xn*B1e6IUP}eo_W*`f){*8oQ=yQeW-&c;#sm&@i_D);lqo?i)tiGibRA zU0x+MOxR}(lurMVz}1puNL?C*B_(i(9X=g}S)+N>4`SHelDa$n(eVw{soT}zef4$1 zqB!xrZwsy$3WZ}&wtszPn7zj1imf$5FZZYP{UqQVdQT7MEcSt(-;TPpP zcfRMu^bugTj8KXmr22<%tYYQA^K~X#az~V1FSxrBchdK2J@)qGM*`6eW@x3XzkmWp z2<^};FDt9@AU%y-inPPxt?Gi9XAViKN|@`9(8`|y__6flA-rD=n!+`1hN>p5%?Riy z+o55IMNidcOX10C+tJZ!cl$VHE zytLtr|Akpe(QnNi#7SWf;nvsnD&2n9#UlJPvR=S0 z5Y@O5-%0aXJ4$=YJ%6gud|pa9kby5*v&8r=&L%e=+<=c{^mal7 z2<(Bjdu^|tdG!bKUP$RpGe=xAe0DS7+up=@eNOz|-VDPmi@3bRpo^!5${C(0#ynyaEu#Beg2YMGm zTXST;F4`U8E&Zl;U(XJKK@RcCXxon-3Ug8T9HUhDfcVY+;o9m2t&^}(vl&ddNP1*B z)x2!Ay(0h&XbE(d@Ez)aKs5d}f5Z(mgRbDqtSv%_CTx+afjJ6a?hw{f%jK`NXG{!4 z(&!spK{|wU#I=zI50!J6R%D%8SBKCz%tZ&CV^VbDvz8y$ZGDfR`Vr{Yjb=V^er-pz z`~e7k#mAJfo+IhEX?-2E_;`h_;0UEU|H`;MI+br}CJOlmP)RTnSp{urHD?2Oe$(fI zsul7qcX$SbxN10|M{+{=*X-GDv!JE#cnpNMF6_qMf5H|S@Qlr615qn&Ww7H3Z0v5< z*+@&kru^{FG3xC~Moy3_l!EL}2RfX2BU?yH$_ZsR_$_W!%uM@jvFmY`PGg+^ODBN) ze+{uYhBXBHY8U%w;j9#0JMS+uz~FE3;(C{!FC;d)>e7yVE0k})#!sUlw3PjqRz=Tz zXfM;;MevjvONNd=!X2>H(A7Gz08uEMgZOoA_Ajn4!YlsYpF0YmX1Zj(@nl84B6W%% z;7TT7%ZEtdYiyp7hGARXVIf$!>h|Dd2>Jt9yM4m23&aq{(9u4pb&Azp1K!LU1-Mtm zJRJrcOlACR{r=Twy^nm%F(B*FGa4?X%202r^>CAMG1#xB(mBfRMdDGO=Dr;wYSr## zb3{fDafgkjIHu3nJ8PRjofXXZcMgS#U32zHh#$WGzF(R<)l!&o^wp*^l=U z?tL4u)sZXf#e?0z5S-m_Pq>J#!Sd@7wlCrq?PV6FiO9JMZZ;IuChuuqr$(bxn$%d1 z-e$68s}0ZJrSN*(_ zaP6;cGjSH%>iC$@WPm}U@YG#a@E9uJtcuGLur@e1Y?(>W@7Y?`V@>7oEmuuVCjt|O3#r#PtDIqQk$-9U=w)lE%z{Fd<8sMEeVs*eu2yo_kk)tOW>M_vhb_S#2zE9+v2j; z`7Up&>`Cv2azFb#0L)hwIii-;Z29S--SGmphfPF`%Z6}FhWf}UCCOx?IOVAXV**rt z!5*HQGKa3<2dcgy^6AZ#yeb~M(nCX|C4T?zt>*Ibu$zD*BZ6hIxP9VzaTjIItC;CNo@&3^7; zc)b+nzpBw{#|Z4@U3dpym}oaihTm!WB+FAt34AaokdF`eke~V)*ZLb5)LYmv1|~(c zYsGofc;0ubtXLyHV&+?IrU&aE2C>*St?XXA4;;EDlcw66TCGm~=4Lgj5I>yZ!|~`3 zM|O-a6JJ z{y0*%I`4IXSC(~DnwXE&>8EFVzHGg`AFq4JYRB}|GeHZOTw4{tZ&+T-Fz)}^b{^`< zdCC@=hkZ}H`SeOje@SgAUA9q}7GCIEm<8_|mT&D1j0L<#5j}}t4`_TolJx&+!Y-XB zggM9<;{)+G4lYjNCyFQ8Ve;_9Z_Zb_WK#RClI#L_&qngUrTSCufu?tD6LuqDZNPI_ zM%_ZS%4^!^mY=^Lm6~}}=WT95ch?c17<7-=?a#Kmiyp3|lRFUiJs{eAENeP9X1e2= zt!1JNeu0}9%8P+SObHnZ?d0_t)8gh~jqq0tQYe7JY)9;&D^y9SK_HbCJ8oEN z`f0nzaTQ^y+s{dc$tiq##0we9_CDS?@8+JM*&B{D4`qclwR6W9Mz&C^pwh$acXe@o z<@7w&m=0{AiX69f!W`q@s{JpwJ{cbwqe1ULdBwZc1AlsJ$H`>JH5L)mKxud}8x5U^n16YlDSed`#1TRsU6aTJ1(h z3$d}p+I0+q^dQ1ICKG4H%!%H}&kfN#sfVURbE*15MVQ9}7n)R|rKmw7UGVt|Q$DEa zVst3Ug~#cut;iFRkq)EKoOfEBrLC7cC|g1d%^P}WU z*dsb}HOy%2(Ut<+R|pPGYj91M;RXwKTnOdh8<>{{2wWzD+bx=%us0Vjh94tJ6S94}MRD=!4HLH^hl!!|9Si!mpz|vfW}eb2BEhTP9-3bB zIGEdW&$EY+%2j>o_14EsjYN<#<6=^f4MFlnb9m*eTzU8r>c7Lw+f1V@oXehE5tmmQRv|O_DaD{igTTEz zz9Yh-vQl{W+{!rYjz+~ZapMek|UaDXn|J`2u zm{TRdg)UW!`g%`dgSfNii8<7*1X8H;Taw@5>NMj(x#1ov!g2Jw!+Bc%{%1o7?ji}D z8Lwlu7?)L@ZzjRN7fCEKFI`#r&URH=&uKB+;+qEIIWQR=SBY) z{)kzg5;87cBRvGscfUIRL)Mk=#5n*7T?|sbI|V4@i2kGFO{L3ztJFgk%#-tTn5nd1 ze9glQgd)R6RafDuF+XrpxktcLAOpOHo!pYVM#SMtpR5mbNaS$DTd>956d>!-C2B#> z3L=^qW!MZoT9o`Y$FV9ZG`sU37bSHmZ}xxn$@09gh&WipAMrnVa_ZaHsL`eN8WI;M z-bq<|bb;X9>fZeE!JNR9t-SECMhdXkdu9FuadhpE2EZg%#|B3;}&MQnGAvRiCM|@8Io#U zl_~7r4#w&DE$&A+U|(aW#}7q+zb{&O4^q=bO{ApeB)jt)3Dm{Ba0um(rDV-AzOv~q zg{BY&vIpx@7&)1IQZLz1A#r!&yu1(6(pfYVAih+uYPBOG<+@47``F6(UHYh;sHtOt z;#{Z=&NFc_;kB?6>!tgjw%HZlWfxT&MjK|^YxgU$dHH%wH#{B`FYd+A^1H@5LtuBk zv9>O}p2)2#DSGW3j+*nKpswqV@R5_}NP5ttQ$8-1Mw+&Wy4#(jPo$PQ%^QQ>#3J>g z1>1+GA(!ahSq&}>Qhe10C->-WM2sd07(+GX)&hf^kk-aWhoI_W=jHyDWQ}1UT;k zq#PYMVso7j)I*t)}uP=u$njy^H(*RCbkNZ9H2)3DzQ^w76C%QXGmq z{Zon*in|97T1p_nDFup_ix;T@#jSXP1ri(zrC5+cafcv9-_ZMhxL@vvd!BuEc6a7? z=IqRz^X$&dnbEO&`fFx0JFbc>O}>imWnL-6$fTXD#AH9i>RIYq$_Kv~>7_d?t$5W# z4Ss7=v@vEsXJgm0fB)5|l%3n}Pks*B-3QLlntBN#Qky9K_?t~M&Fv=!;Jo2}?TMz< zq4uZ}xWfS+IRdgV1|s=Luu-!glc$%$x`6SE=dn5DN;v-rzs)&2#$h#3#r`CPhh@ zgQ8gXTAX6Om2?(xSx-AV#@UHM!{UOdk#rYOLXQ<}Vbj(^$AAUBO|aV3WHa7~`Yj#k z=5}juEj4CTg}%Etyflqq*5-)jm`?^fBrzpY=QCs_EByhKX_`t-!sUG}n0==W4%WJ0 z$$F*}@;*J*G_yWEvK@JD%u=9;x^Vb#5<>4f{{ zcN|Q@osxx9w@IX!8ZkN8A#{IgOC~T?LD!$}NHFa!%_Jp~Ub@%bF9AA~n0L=&@Mt&$$ zr2xxh+QTxxouE^!A3U9GuaPVOj=m}P6h^@h?B;bzhL_w+xptusTr7Sz9@Mp) zNzP(cJ4_h91Z*s(*UY8=GV>D3yc}3VcO)&sNz;RtaXOHM*EpI462!!zcvLH5KDj-A z!r$6hOJN9&53DX7{@C_0vdRUUjLVtmiw(#Xh?BPPlQUS>U(;TDHc(ciFghS~G%;kG zQFTaq91wC8TavrV$g@&v(v+m!9qiPe!{75XvqXem>|9{-Pn%@S)_XojM~SF1&2)ss z!TBV&;gy`Bc;!N?AWo41mv1wMW;37GS(ED9C=PMn4w3r3C?7{$YlFF19xHHpXP%COOt$$o53F7REj=GSo&dsbOJ4>+xTm%r7 z`Je9yYo<4m0H4M2CQg2P&(#@56?qpCftGcIe#@X@wDeA*L~?Xj9{uQB&G{Nm(VRMwG$eJg+_Zl8n3rOc(k5*2fqX(> zTYtIP#k{14W#TiQ@l~+qDam5;}YBPWS!2(V?8uOahd|<7aza{Ws;7NdppA}+<#q;G)D$SX# zcVRq%#o5N2jO!DSwVL;#?}Jmn1r;(ZyJcg#b>sykav*u5C@(rz?yy(XyJ=`f}M&5d&O0;ss4R9X+`Zc6ML0y3r0=(@R zgKb2k8OiWxUyv7HkNYP&b@?TavC=U#{`t*s5^ti=sl{DCu^i(>C3UCq4)lW?^^LeRfjWQLAj~ zGgnl*7+6PE?ohY#ts6C~2!mnd~CP{gFp!eHtEA^x#q%(x zFKgK>%DBfk1)47_NS88nP}*|@5zv-)B$NhjqmR4!DMd2NzRL7YDB0s*CeT^uHvu;K zp~X^6R}*o%OJMOJ1*?2M?Ir)BrgW@hD~;l1ro(p+#Ipxq>uGXJX*q#VDV2L3Q{JU= zjXY2unI!Q#x6t9$$m!EG+@SW3|8a+`>>LYu#rjR24OClfWXbUgTW|mfDZ}*W*C*B3 zsdRqR5oz|;vuJ&p^%`;U=MlXS{oLyw%{|)xA(h6^m%E2AhNGgQ*w7 zREAqL0UlxQ*sY73cN~A;o61*{tFzIho$%n$x2SczEQL;QSaNXL=I98>G`60#@hwWd z>a_ZncwvG!M~`QzZiEZBrg_=8n~c?$H*KS+1_nGmirhl{`VnOt#upyuJ4mOWT4|yu zT_KC}Om0$DGJW4|a}!Bv)FyZN5OMXTEYp7sG@ zCi6yrkFw51@M@)W7OHueJ?bdn$DVZ&!L1hL#GMnqBTL(m_>Nm(vroaES;^W9oqa%k z7JOE%hjNi){K7=}*dkc2)3FhnF)5c+JoV#!`Pyt+`OLYM;gjQntb&0(e=0FyX78hb z-_K;CnSHz)jO(`rJ`Cn){R(lxZV-u~Q9U1-3TZya&o{xcmUCHx_q&7x zgRcDM^mi87$J4gQopu0`m~|5v9pOzU#Bj?SsnUBAaRJCWldH8_y4L*<67$@$ zH?xZ2aBOKqR`F+#u;x8@fnxvAW_E7L>|^=7LG44Y%1mL!F?1ls$>aL3T_8`tg(43H zDrC9dzQB!_S}sAUyk$LKOWIqw>O5WTZQS@S%*~i4Ml=-63qf3Z9U20m+uWIZI7LpS zZI;C#KZ_XOgf0%$?&*_n1^L~p_+R*jD(TRATp*Q)+W@po)fdhM4?4UUFXw0)6~wizM5s++SyvY>%R?PMi)ruSjN8+T7VZ6%a6(WqAo|K;~;=3AMV?U>On7`d@-z+a5OyJI)*wJ~!UkiX! z^-P6T(7C@SLW*O~;)PZn)F78h{9Uq=~2tmmNh>C3p0`d^}6RQOfCqT8Da|FC~&J=*fC-r@}|r z0;XZdmT{I7gdVEhiH2zP7=nn-Q z6L{HMf4qaUUcO`;vuv)voXQi(@hW)!?Y-$1FirwQ<_6Qi@f+@tVLE z-bgRYkgz~9;btcNq~ZtsLo0kZ>Hn(5c)`&DT^D-YRG1VLV8K#W#~jh!ODT-o&^jNI6E*=)px0Z4KSGt62U zi3i9Cc+nQyCQbJup%4g5buqn^+a*sTF#<8HniRJb%m4giw3njwEssde$a#tyI4f{K z9>lS0_~GU=A`T!s=_OKYw3+sVWnSq6xfG>8al_Ys9Su)VY+ogR9-x#+8^|2`uQ0-n z!!l{b6|RY|S`6hL-f@v0HVRwQCQ1b52_BGNm@1sue9CM)6@9)#5_!kFN@9LXQ~}dR zj!hBe?}GtDL?F|4^K-BG4Xwx6Hv&uiHMmBKJI1{78-Wd_;o1gWjzA&97jW}lxllnP z4~IX7HkV26QJ~iERR%$Zh|+_y1>VeOGi{S?Vx0EXR=^g6Wfuh!|F=0hex+aT7_#)Z zcM2K2$hs;T4&@6Klt(EIB)QkNNw8^-8P{TE<{srH??FS=;9I){e{CDm9^$Cc(D7%O zg;aa^`>=6!c$LP*KjK+<@bLi*q)hT9U8Ms1S?LuC+oI2HDt#lOyA^;Y{#1C-OnQ@L z@z$Pm%IF`z$GyC{$w9uimt@`WR`V5n_`(m z{zHtO0`>{3W#bDps>Sjiwmt^Jl<%WGM;%u3sB$}Px@V0pdGf3aOcq>bUvyf0WU^x3 zr7!x< z`dM?Cr%!$*RT8F?ELS?OSv>4u%d0*#2u)u41O~E$S}Y3IP;@2+?T3xH-hEw8KGAHA zYB1g<@`|JQ-+3>Y5t_^wqvUBPPofA;qS-40Wwl9X;!6_$La#t;(4fE~{%Krb-)^_4 zU@Jnlu{tIDc{S8B4YG=1a+sx0J*X)cmK!jZgIoi=|9TD{Vh_{*GAP>?)j=+RWLuI;puO-r@^iT;FjmpT$4 z^Hg*EuTYzxm`^;D^VbLYpl%OLgV15-DXP~Re?NvaSv^8zSM7-c;_EFXPmpVh;q_y^ zwe>J33g!Jd>VrQ4n-=klDh}z_g=Wxn#ZR8}KI)GgNhx38Wp=gZwFr+aVY+~ziyMkB zEx!+yatG*r{ND<=yWupXNex-Q5ysN~1K{onad8az?|_g{{4xOmP*>Idr(DG*{Qm&Y%4*F3 literal 0 HcmV?d00001 diff --git a/src/statistical-and-mathematical-methods-for-ai/main.tex b/src/statistical-and-mathematical-methods-for-ai/main.tex index 2356a8b..607f7ab 100644 --- a/src/statistical-and-mathematical-methods-for-ai/main.tex +++ b/src/statistical-and-mathematical-methods-for-ai/main.tex @@ -13,5 +13,6 @@ \input{sections/_matrix_decomp.tex} \input{sections/_vector_calculus.tex} \input{sections/_gradient_methods.tex} + \input{sections/_probability.tex} \end{document} \ No newline at end of file diff --git a/src/statistical-and-mathematical-methods-for-ai/sections/_probability.tex b/src/statistical-and-mathematical-methods-for-ai/sections/_probability.tex new file mode 100644 index 0000000..5498d57 --- /dev/null +++ b/src/statistical-and-mathematical-methods-for-ai/sections/_probability.tex @@ -0,0 +1,210 @@ +\chapter{Probability} + + +\section{Probability} +\begin{description} + \item[State space] \marginnote{State space} + Set $\Omega$ of all the possible results of an experiment. + \begin{example} + A coin is tossed two times. + $\Omega = \{ (\text{T}, \text{T}), (\text{T}, \text{H}), (\text{H}, \text{T}), (\text{H}, \text{H}) \}$ + \end{example} + + \item[Event] \marginnote{Event} + Set of possible results (i.e. $A$ is an event if $A \subseteq \Omega$) + + \item[Probability] \marginnote{Probability} + Let $\mathbb{E}$ be the set of all the possible events (i.e. power set of $\Omega$). + The probability is a function: + \[ \prob{A}: \mathbb{E} \rightarrow [0, 1] \] + \begin{example} + Let $\Omega$ be as above. + Given an event $A = \{ (\text{T}, \text{H}), (\text{H}, \text{T}) \}$, + its probability is: $\prob{A} = \frac{2}{4} = \frac{1}{2}$ + \end{example} + + \item[Conditional probability] \marginnote{Conditional probability} + Probability of an event $B$, knowing that another event $A$ happened: + \[ \prob{B \vert A} = \frac{\prob{A \cap B}}{\prob{A}} \text{, with } \prob{A} \neq 0 \] + + \begin{example} + A coin is tossed three times. + Given the events $A = \{ \text{tails two times} \}$ and $B = \{ \text{one heads and one tails} \}$ + We have that: + + \begin{minipage}{\linewidth} + \centering + \small + $\Omega = \{ + (\text{T}, \text{T}, \text{T}), (\text{T}, \text{T}, \text{H}), (\text{T}, \text{H}, \text{T}) + (\text{T}, \text{H}, \text{H}), (\text{H}, \text{T}, \text{T}), (\text{H}, \text{T}, \text{H}) + (\text{H}, \text{H}, \text{T}), (\text{H}, \text{H}, \text{H}) + \}$ + \end{minipage} + + \begin{minipage}{.325\linewidth} + \centering + $\prob{A} = \frac{4}{8} = \frac{1}{2}$ + \end{minipage} + \begin{minipage}{.325\linewidth} + \centering + $\prob{B} = \frac{6}{8} = \frac{3}{4}$ + \end{minipage} + \begin{minipage}{.325\linewidth} + \centering + $\prob{A \cap B} = \frac{3}{8}$ + \end{minipage} + + \begin{minipage}{.48\linewidth} + \centering + $\prob{A \vert B} = \frac{3/8}{3/4} = \frac{1}{2}$ + \end{minipage} + \begin{minipage}{.48\linewidth} + \centering + $\prob{B \vert A} = \frac{3/8}{1/2} = \frac{3}{4}$ + \end{minipage} + \end{example} + + \item[Independent events] \marginnote{Independent events} + Two events $A$ and $B$ are independent if: + \[ \prob{A \cap B} = \prob{A}\prob{B} \] + It follows that: + + \begin{minipage}{.48\linewidth} + \centering + $\prob{A \vert B} = \prob{A}$ + \end{minipage} + \begin{minipage}{.48\linewidth} + \centering + $\prob{B \vert A} = \prob{B}$ + \end{minipage} + + In general, given $n$ events $A_1, \dots, A_n$, they are independent if: + \[ \prob{A_1 \cap \dots \cap A_n} = \prod_{i=1}^{n} \prob{A_i} \] +\end{description} + + + +\section{Random variables} +\begin{description} + \item[Random variable (RV)] \marginnote{Random variable} + A random variable $X$ is a function: + \[ X: \Omega \rightarrow \mathbb{R} \] + + \item[Target space/Support] \marginnote{Target space} + Given a random variable $X$, + the target space (or support) $\mathcal{T}_X$ of $X$ is the set of all its possible values: + \[ \mathcal{T}_X = \{ x \mid x = X(\omega), \forall \omega \in \Omega \} \] +\end{description} + + +\subsection{Discrete random variables} + +\begin{description} + \item[Discrete random variable] \marginnote{Discrete random variable} + A random variable $X$ is discrete if its target space $\mathcal{T}_X$ is finite or countably infinite. + + \begin{example} + A coin is tossed twice. + + The random variable is $X(\omega) = \{ \text{number of heads} \}$. + We have that $\mathcal{T}_X = \{ 0, 1, 2 \}$, therefore $X$ is discrete. + \end{example} + + \begin{example} + Roll a die until 6 comes out. + + The random variable is $Y(\omega) = \{ \text{number of rolls before 6} \}$. + We have that $\mathcal{T}_Y = \{ 1, 2, \dots \} = \mathbb{N} \smallsetminus \{0\}$, + therefore $Y$ is discrete as $\mathcal{T}_Y$ is a countable set. + \end{example} + + \item[Probability mass function (PMF)] \marginnote{Probability mass function (PMF)} + Given a discrete random variable $X$, its probability mass function is a function $f_X: \mathcal{T}_X \rightarrow [0, 1]$ such that: + \[ f_X(x) = \prob{X = x}, \forall x \in \mathcal{T}_X \] + + A PMF has the following properties: + \begin{enumerate} + \item $f_X(x) \geq 0, \forall x \in \mathcal{T}_X$ + \item $\sum_{x \in \mathcal{T}_X} f_X(x) = 1$ + \item Let $A \subseteq \Omega$, $\prob{X = x \in A} = \sum_{x \in A} f_X(x)$ + \end{enumerate} + + \begin{example} + Let $\Omega = \{ (\text{T}, \text{T}), (\text{T}, \text{H}), (\text{H}, \text{T}), (\text{H}, \text{H}) \}$. + Given a random variable $X = \{ \text{number of heads} \}$ with $\mathcal{T}_X = \{ 0, 1, 2 \}$. + The PMF is: + \[ + \begin{split} + f_X &= \prob{X = 0} = \frac{1}{4} \\ + f_X &= \prob{X = 1} = \frac{2}{4} \\ + f_X &= \prob{X = 2} = \frac{1}{4} + \end{split} + \] + \end{example} +\end{description} + +\subsubsection{Common distributions} +\begin{descriptionlist} + \item[Uniform distribution] \marginnote{Uniform distribution} + Given a discrete random variable $X$ with $\#(\mathcal{T}_X) = N$, + $X$ has an uniform distribution if: + \[ f_X(x) = \frac{1}{N}, \forall x \in \mathcal{T}_X \] + + \item[Poisson distribution] \marginnote{Poisson distribution} + Given a discrete random variable $X$ with mean $\lambda$, + $X$ has a poisson distribution if: + \[ f_X(x) = e^{-\lambda} \frac{\lambda^x}{x!}, \forall x \in \mathcal{T}_X \] +\end{descriptionlist} + + +\subsection{Continuous random variables} + +\begin{description} + \item[Continuous random variable] \marginnote{Continuous random variable} + A random variable $X$ is continuous if its target space $\mathcal{T}_X$ is uncountably infinite (i.e. a subset of $\mathbb{R}$). + Usually, $\mathcal{T}_X$ is an interval or union of intervals. + + \begin{example} + Given a random variable $Z = \{ \text{Time before the arrival of a client} \}$. + $Z$ is continuous as $\mathcal{T}_Z = [a, b] \subseteq [0, +\infty[$ is an uncountable set. + \end{example} + + \item[Probability density function (PDF)] \marginnote{Probability density function (PDF)} + Given a continuous random variable $X$, + its probability density function is a function $f_X: \mathcal{T}_X \rightarrow \mathbb{R}$ such that: + \[ \prob{X \in A} = \int_{A} f_X(x) \,dx \] + \[ \prob{a \leq X \leq b} = \int_{a}^{b} f_X(x) \,dx \] + Note that $\prob{X = a} = \prob{a \leq X \leq a} = \int_{a}^{a} f_X(x) \,dx = 0$ + + A PDF has the following properties: + \begin{enumerate} + \item $f_X(x) \geq 0, \forall x \in \mathcal{T}_X$ + \item $\int_{x \in \mathcal{T}_X} f_X(x) \,dx = 1$ + \item $\prob{X \in A} = \int_{A} f_X(x) \,dx$ + \end{enumerate} +\end{description} + +\subsubsection{Common distributions} +\begin{descriptionlist} + \item[Continuous uniform distribution] \marginnote{Continuous uniform distribution} + Given a continuous random variable $X$ with $\mathcal{T}_X = [a, b]$, + $X$ has a continuous uniform distribution if: + \[ f_X(x) = \frac{1}{b-a}, \forall x \in \mathcal{T}_X \] + + \item[Normal distribution] \marginnote{Normal distribution} + Given a continuous random variable $X$ and the parameters $\mu$ (mean) and $\sigma$ (variance). + $X$ has a normal distribution if: + \[ f_X(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{\frac{-(x-\mu)^2}{2\sigma^2}} , \forall x \in \mathcal{T}_X\] + + \begin{description} + \item[Standard normal distribution] \marginnote{Standard normal distribution} + Normal distribution with $\mu = 0$ and $\sigma = 1$. + \end{description} + + \begin{figure}[ht] + \centering + \includegraphics[width=0.5\textwidth]{img/normal_distribution.png} + \caption{Normal distributions and standard normal distribution} + \end{figure} +\end{descriptionlist} \ No newline at end of file