diff options
author | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
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committer | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
commit | e0c6872cf40896c7be36b11dcc744620f10adf1d (patch) | |
tree | 60335e10d2f4354b0674ec22d7b53f0f8abee672 /macros/latex/contrib/pbsheet/xpl |
Initial commit
Diffstat (limited to 'macros/latex/contrib/pbsheet/xpl')
-rw-r--r-- | macros/latex/contrib/pbsheet/xpl/GNUmakefile | 85 | ||||
-rw-r--r-- | macros/latex/contrib/pbsheet/xpl/img/simbin.eps | 2459 | ||||
-rw-r--r-- | macros/latex/contrib/pbsheet/xpl/img/simbin.pdf | bin | 0 -> 12612 bytes | |||
-rw-r--r-- | macros/latex/contrib/pbsheet/xpl/pbsheet.cls | 551 | ||||
-rw-r--r-- | macros/latex/contrib/pbsheet/xpl/pgm/probadis.m | 65 | ||||
-rw-r--r-- | macros/latex/contrib/pbsheet/xpl/pgm/rdiscr.m | 72 | ||||
-rw-r--r-- | macros/latex/contrib/pbsheet/xpl/pgm/rint.m | 25 | ||||
-rw-r--r-- | macros/latex/contrib/pbsheet/xpl/pgm/simbin.m | 55 | ||||
-rw-r--r-- | macros/latex/contrib/pbsheet/xpl/xpl-fr.bib | 39 | ||||
-rw-r--r-- | macros/latex/contrib/pbsheet/xpl/xpl-fr.dvi | bin | 0 -> 185352 bytes | |||
-rw-r--r-- | macros/latex/contrib/pbsheet/xpl/xpl-fr.pdf | bin | 0 -> 255740 bytes | |||
-rw-r--r-- | macros/latex/contrib/pbsheet/xpl/xpl-fr.ps | 7998 | ||||
-rw-r--r-- | macros/latex/contrib/pbsheet/xpl/xpl-fr.tex | 146 |
13 files changed, 11495 insertions, 0 deletions
diff --git a/macros/latex/contrib/pbsheet/xpl/GNUmakefile b/macros/latex/contrib/pbsheet/xpl/GNUmakefile new file mode 100644 index 0000000000..035b81b336 --- /dev/null +++ b/macros/latex/contrib/pbsheet/xpl/GNUmakefile @@ -0,0 +1,85 @@ +# GNUmakefile, © 2003, 2004, Djalil CHAFAI. + +# Please adjust the following variables to your environment. + +BIBTEX=bibtex +DVIPS=dvips -D 600 -t a4 +LATEX=latex #--src-specials +PDFLATEX=pdflatex +RM=rm +THUMBPDF=thumbpdf + +# Normally, what follows does not need any modification. + +NOM=xpl-fr +NOML="" +MAKEFILENAME=GNUmakefile +OUTPUTS=dvi ps pdf +INTERMED=log out aux toc ind bbl blg lol lof lot ilg idx inc rel brf tpt +FIGSDIR=img +CODESDIR=pgm +TEXINPUTS:=.:./$(FIGSDIR)//:./$(CODESDIR)//:$(TEXINPUTS) +export TEXINPUTS + +# Static sources related variables. + +TEX= $(NOM).bib $(NOM).tex +MISC1= $(shell kpsewhich amsalpha.bst) +MISC2= $(wildcard $(FIGSDIR)/* $(CODESDIR)/*) $(MAKEFILENAME) +SRC= $(MISC1) $(MISC2) $(TEX) + +# Dependencies. + +DEP_PS=$(MAKEFILENAME) $(MISC1) $(TEX) +DEP_PDF=$(MAKEFILENAME) $(MISC1) $(TEX) +DEP_ARCH=$(SRC) $(NOM).ps $(NOM).pdf + + +# Special variables. + +.PHONY= $(OUTPUTS) help clean cleanall +.PHONY: $(.PHONY) +.SUFFIXES: +.INTERMEDIATE: $(INTERMDED:%=$(NOM).%) + +# Phony rules. + +help: + @echo + @echo " Rules to compile $(NOM).tex : $(.PHONY)." + @echo " For example «make ps» will make the PS file for you!" + @echo + +dvi: clean $(NOM).dvi +ps: $(NOM).ps +pdf: clean $(NOM).pdf + +clean: + @$(RM) -f $(INTERMED:%=$(NOM).%) thb*.png *~ *.aux *.inc + @$(RM) -f $(CODESDIR)/*~ $(FIGSDIR)/*~ + @$(RM) -rf auto + +cleanall: clean + @for f in $(OUTPUTS); do $(RM) -f $(NOM).$$f; done + +# Rules based on target files. + +$(NOM).ps: $(NOM).dvi + @$(DVIPS) $(NOM).dvi -o $(NOM).ps + +$(NOM).dvi: $(DEP_PS) + @$(LATEX) $(NOM) + @$(BIBTEX) $(NOM) + @$(LATEX) $(NOM) + @$(LATEX) $(NOM) + @$(LATEX) $(NOM) + +$(NOM).pdf: $(DEP_PDF) + @$(PDFLATEX) $(NOM) + @$(BIBTEX) $(NOM) + @$(PDFLATEX) $(NOM) + @$(PDFLATEX) $(NOM) + @$(THUMBPDF) $(NOM).pdf + @$(PDFLATEX) $(NOM) + +# End Of File. diff --git a/macros/latex/contrib/pbsheet/xpl/img/simbin.eps b/macros/latex/contrib/pbsheet/xpl/img/simbin.eps new file mode 100644 index 0000000000..6403e0671f --- /dev/null +++ b/macros/latex/contrib/pbsheet/xpl/img/simbin.eps @@ -0,0 +1,2459 @@ +%!PS-Adobe-2.0 EPSF-2.0 +%%Title: gauss.eps +%%Creator: gnuplot 3.7 patchlevel 3 +%%CreationDate: Mon Dec 8 17:23:19 2003 +%%DocumentFonts: (atend) +%%Orientation: Portrait +%%BoundingBox: 0 44 300 255 +%%EndComments +%%BeginProcSet: epsffit 1 0 +gsave +-41.667 3.333 translate +0.833 0.833 scale +%%EndProcSet + +/gnudict 256 dict def +gnudict begin +/Color true def +/Solid true def +/gnulinewidth 5.000 def +/userlinewidth gnulinewidth def +/vshift -46 def +/dl {10 mul} def +/hpt_ 31.5 def +/vpt_ 31.5 def +/hpt hpt_ def +/vpt vpt_ def +/M {moveto} bind def +/L {lineto} bind def +/R {rmoveto} bind def +/V {rlineto} bind def +/vpt2 vpt 2 mul def +/hpt2 hpt 2 mul def +/Lshow { currentpoint stroke M + 0 vshift R show } def +/Rshow { currentpoint stroke M + dup stringwidth pop neg vshift R show } def +/Cshow { currentpoint stroke M + dup stringwidth pop -2 div vshift R show } def +/UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def + /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def +/DL { Color {setrgbcolor Solid {pop []} if 0 setdash } + {pop pop pop Solid {pop []} if 0 setdash} ifelse } def +/BL { stroke userlinewidth 2 mul setlinewidth } def +/AL { stroke userlinewidth 2 div setlinewidth } def +/UL { dup gnulinewidth mul /userlinewidth exch def + dup 1 lt {pop 1} if 10 mul /udl exch def } def +/PL { stroke userlinewidth setlinewidth } def +/LTb { BL [] 0 0 0 DL } def +/LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def +/LT0 { PL [] 1 0 0 DL } def +/LT1 { PL [4 dl 2 dl] 0 1 0 DL } def +/LT2 { PL [2 dl 3 dl] 0 0 1 DL } def +/LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def +/LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def +/LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def +/LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def +/LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def +/LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def +/Pnt { stroke [] 0 setdash + gsave 1 setlinecap M 0 0 V stroke grestore } def +/Dia { stroke [] 0 setdash 2 copy vpt add M + hpt neg vpt neg V hpt vpt neg V + hpt vpt V hpt neg vpt V closepath stroke + Pnt } def +/Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V + currentpoint stroke M + hpt neg vpt neg R hpt2 0 V stroke + } def +/Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M + 0 vpt2 neg V hpt2 0 V 0 vpt2 V + hpt2 neg 0 V closepath stroke + Pnt } def +/Crs { stroke [] 0 setdash exch hpt sub exch vpt add M + hpt2 vpt2 neg V currentpoint stroke M + hpt2 neg 0 R hpt2 vpt2 V stroke } def +/TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M + hpt neg vpt -1.62 mul V + hpt 2 mul 0 V + hpt neg vpt 1.62 mul V closepath stroke + Pnt } def +/Star { 2 copy Pls Crs } def +/BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M + 0 vpt2 neg V hpt2 0 V 0 vpt2 V + hpt2 neg 0 V closepath fill } def +/TriUF { stroke [] 0 setdash vpt 1.12 mul add M + hpt neg vpt -1.62 mul V + hpt 2 mul 0 V + hpt neg vpt 1.62 mul V closepath fill } def +/TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M + hpt neg vpt 1.62 mul V + hpt 2 mul 0 V + hpt neg vpt -1.62 mul V closepath stroke + Pnt } def +/TriDF { stroke [] 0 setdash vpt 1.12 mul sub M + hpt neg vpt 1.62 mul V + hpt 2 mul 0 V + hpt neg vpt -1.62 mul V closepath fill} def +/DiaF { stroke [] 0 setdash vpt add M + hpt neg vpt neg V hpt vpt neg V + hpt vpt V hpt neg vpt V closepath fill } def +/Pent { stroke [] 0 setdash 2 copy gsave + translate 0 hpt M 4 {72 rotate 0 hpt L} repeat + closepath stroke grestore Pnt } def +/PentF { stroke [] 0 setdash gsave + translate 0 hpt M 4 {72 rotate 0 hpt L} repeat + closepath fill grestore } def +/Circle { stroke [] 0 setdash 2 copy + hpt 0 360 arc stroke Pnt } def +/CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def +/C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def +/C1 { BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 90 arc closepath fill + vpt 0 360 arc closepath } bind def +/C2 { BL [] 0 setdash 2 copy moveto + 2 copy vpt 90 180 arc closepath fill + vpt 0 360 arc closepath } bind def +/C3 { BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 180 arc closepath fill + vpt 0 360 arc closepath } bind def +/C4 { BL [] 0 setdash 2 copy moveto + 2 copy vpt 180 270 arc closepath fill + vpt 0 360 arc closepath } bind def +/C5 { BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 90 arc + 2 copy moveto + 2 copy vpt 180 270 arc closepath fill + vpt 0 360 arc } bind def +/C6 { BL [] 0 setdash 2 copy moveto + 2 copy vpt 90 270 arc closepath fill + vpt 0 360 arc closepath } bind def +/C7 { BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 270 arc closepath fill + vpt 0 360 arc closepath } bind def +/C8 { BL [] 0 setdash 2 copy moveto + 2 copy vpt 270 360 arc closepath fill + vpt 0 360 arc closepath } bind def +/C9 { BL [] 0 setdash 2 copy moveto + 2 copy vpt 270 450 arc closepath fill + vpt 0 360 arc closepath } bind def +/C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill + 2 copy moveto + 2 copy vpt 90 180 arc closepath fill + vpt 0 360 arc closepath } bind def +/C11 { BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 180 arc closepath fill + 2 copy moveto + 2 copy vpt 270 360 arc closepath fill + vpt 0 360 arc closepath } bind def +/C12 { BL [] 0 setdash 2 copy moveto + 2 copy vpt 180 360 arc closepath fill + vpt 0 360 arc closepath } bind def +/C13 { BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 90 arc closepath fill + 2 copy moveto + 2 copy vpt 180 360 arc closepath fill + vpt 0 360 arc closepath } bind def +/C14 { BL [] 0 setdash 2 copy moveto + 2 copy vpt 90 360 arc closepath fill + vpt 0 360 arc } bind def +/C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill + vpt 0 360 arc closepath } bind def +/Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto + neg 0 rlineto closepath } bind def +/Square { dup Rec } bind def +/Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def +/S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def +/S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def +/S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def +/S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def +/S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def +/S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill + exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def +/S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def +/S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill + 2 copy vpt Square fill + Bsquare } bind def +/S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def +/S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def +/S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill + Bsquare } bind def +/S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill + Bsquare } bind def +/S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def +/S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill + 2 copy vpt Square fill Bsquare } bind def +/S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill + 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def +/S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def +/D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def +/D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def +/D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def +/D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def +/D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def +/D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def +/D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def +/D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def +/D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def +/D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def +/D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def +/D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def +/D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def +/D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def +/D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def +/D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def +/DiaE { stroke [] 0 setdash vpt add M + hpt neg vpt neg V hpt vpt neg V + hpt vpt V hpt neg vpt V closepath stroke } def +/BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M + 0 vpt2 neg V hpt2 0 V 0 vpt2 V + hpt2 neg 0 V closepath stroke } def +/TriUE { stroke [] 0 setdash vpt 1.12 mul add M + hpt neg vpt -1.62 mul V + hpt 2 mul 0 V + hpt neg vpt 1.62 mul V closepath stroke } def +/TriDE { stroke [] 0 setdash vpt 1.12 mul sub M + hpt neg vpt 1.62 mul V + hpt 2 mul 0 V + hpt neg vpt -1.62 mul V closepath stroke } def +/PentE { stroke [] 0 setdash gsave + translate 0 hpt M 4 {72 rotate 0 hpt L} repeat + closepath stroke grestore } def +/CircE { stroke [] 0 setdash + hpt 0 360 arc stroke } def +/Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def +/DiaW { stroke [] 0 setdash vpt add M + hpt neg vpt neg V hpt vpt neg V + hpt vpt V hpt neg vpt V Opaque stroke } def +/BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M + 0 vpt2 neg V hpt2 0 V 0 vpt2 V + hpt2 neg 0 V Opaque stroke } def +/TriUW { stroke [] 0 setdash vpt 1.12 mul add M + hpt neg vpt -1.62 mul V + hpt 2 mul 0 V + hpt neg vpt 1.62 mul V Opaque stroke } def +/TriDW { stroke [] 0 setdash vpt 1.12 mul sub M + hpt neg vpt 1.62 mul V + hpt 2 mul 0 V + hpt neg vpt -1.62 mul V Opaque stroke } def +/PentW { stroke [] 0 setdash gsave + translate 0 hpt M 4 {72 rotate 0 hpt L} repeat + Opaque stroke grestore } def +/CircW { stroke [] 0 setdash + hpt 0 360 arc Opaque stroke } def +/BoxFill { gsave Rec 1 setgray fill grestore } def +/Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont +dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall +currentdict end definefont pop +/MFshow {{dup dup 0 get findfont exch 1 get scalefont setfont + [ currentpoint ] exch dup 2 get 0 exch rmoveto dup dup 5 get exch 4 get + {show} {stringwidth pop 0 rmoveto}ifelse dup 3 get + {2 get neg 0 exch rmoveto pop} {pop aload pop moveto}ifelse} forall} bind def +/MFwidth {0 exch {dup 3 get{dup dup 0 get findfont exch 1 get scalefont setfont + 5 get stringwidth pop add} + {pop} ifelse} forall} bind def +/MLshow { currentpoint stroke M + 0 exch R MFshow } bind def +/MRshow { currentpoint stroke M + exch dup MFwidth neg 3 -1 roll R MFshow } def +/MCshow { currentpoint stroke M + exch dup MFwidth -2 div 3 -1 roll R MFshow } def +end +%%EndProlog +gnudict begin +gsave +50 50 translate +0.050 0.050 scale +0 setgray +newpath +(Helvetica) findfont 140 scalefont setfont +1.000 UL +LTb +714 420 M +63 0 V +6185 0 R +-63 0 V + stroke +630 420 M +[ [(Helvetica) 140.0 0.0 true true ( 0.8)] +] -46.7 MRshow +714 1120 M +63 0 V +6185 0 R +-63 0 V + stroke +630 1120 M +[ [(Helvetica) 140.0 0.0 true true ( 1)] +] -46.7 MRshow +714 1820 M +63 0 V +6185 0 R +-63 0 V + stroke +630 1820 M +[ [(Helvetica) 140.0 0.0 true true ( 1.2)] +] -46.7 MRshow +714 2520 M +63 0 V +6185 0 R +-63 0 V + stroke +630 2520 M +[ [(Helvetica) 140.0 0.0 true true ( 1.4)] +] -46.7 MRshow +714 3220 M +63 0 V +6185 0 R +-63 0 V + stroke +630 3220 M +[ [(Helvetica) 140.0 0.0 true true ( 1.6)] +] -46.7 MRshow +714 3920 M +63 0 V +6185 0 R +-63 0 V + stroke +630 3920 M +[ [(Helvetica) 140.0 0.0 true true ( 1.8)] +] -46.7 MRshow +714 4620 M +63 0 V +6185 0 R +-63 0 V + stroke +630 4620 M +[ [(Helvetica) 140.0 0.0 true true ( 2)] +] -46.7 MRshow +714 420 M +0 63 V +0 4137 R +0 -63 V + stroke +714 280 M +[ [(Helvetica) 140.0 0.0 true true ( 0)] +] -46.7 MCshow +1339 420 M +0 63 V +0 4137 R +0 -63 V + stroke +1339 280 M +[ [(Helvetica) 140.0 0.0 true true ( 100)] +] -46.7 MCshow +1964 420 M +0 63 V +0 4137 R +0 -63 V + stroke +1964 280 M +[ [(Helvetica) 140.0 0.0 true true ( 200)] +] -46.7 MCshow +2588 420 M +0 63 V +0 4137 R +0 -63 V + stroke +2588 280 M +[ [(Helvetica) 140.0 0.0 true true ( 300)] +] -46.7 MCshow +3213 420 M +0 63 V +0 4137 R +0 -63 V + stroke +3213 280 M +[ [(Helvetica) 140.0 0.0 true true ( 400)] +] -46.7 MCshow +3838 420 M +0 63 V +0 4137 R +0 -63 V + stroke +3838 280 M +[ [(Helvetica) 140.0 0.0 true true ( 500)] +] -46.7 MCshow +4463 420 M +0 63 V +0 4137 R +0 -63 V + stroke +4463 280 M +[ [(Helvetica) 140.0 0.0 true true ( 600)] +] -46.7 MCshow +5088 420 M +0 63 V +0 4137 R +0 -63 V + stroke +5088 280 M +[ [(Helvetica) 140.0 0.0 true true ( 700)] +] -46.7 MCshow +5712 420 M +0 63 V +0 4137 R +0 -63 V + stroke +5712 280 M +[ [(Helvetica) 140.0 0.0 true true ( 800)] +] -46.7 MCshow +6337 420 M +0 63 V +0 4137 R +0 -63 V + stroke +6337 280 M +[ [(Helvetica) 140.0 0.0 true true ( 900)] +] -46.7 MCshow +6962 420 M +0 63 V +0 4137 R +0 -63 V + stroke +6962 280 M +[ [(Helvetica) 140.0 0.0 true true ( 1000)] +] -46.7 MCshow +1.000 UL +LTb +714 420 M +6248 0 V +0 4200 V +-6248 0 V +714 420 L + stroke +140 2520 M +currentpoint gsave translate 90 rotate 0 0 moveto +[ [(Helvetica) 140.0 0.0 true true (Moyennes empiriques)] +] -46.7 MCshow +grestore +3838 70 M +[ [(Helvetica) 140.0 0.0 true true (Nombre de realisations)] +] -46.7 MCshow +3838 4830 M +[ [(Helvetica) 140.0 0.0 true true (Loi des Grands Nombres)] +] -46.7 MCshow +1.000 UL +LT2 +6311 4487 M +[ [(Helvetica) 140.0 0.0 true true (Empirique)] +] -46.7 MRshow +6395 4487 M +399 0 V +720 4620 M +6 -3500 V +7 1167 V +6 -292 V +6 1225 V +6 233 V 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license +%% or (at your option) any later version. The latest version of this +%% license is in: +%% +%% http://www.latex-project.org/lppl.txt +%% +%% and version 1.2 or later is part of all distributions of LaTeX version +%% 1999/12/01 or later. +%% +\NeedsTeXFormat{LaTeX2e}[1999/12/01] +\ProvidesClass{pbsheet} + [2004/02/14 v0.1 .dtx pbsheet file] +\ClassInfo{pbsheet}{Copyright (c) Djalil CHAFAI, 2003, 2004.} +\ClassInfo{pbsheet}{http://www.lsp.ups-tlse.fr/Chafai/} +\ClassInfo{pbsheet}{chafai AT math DOT ups-tlse DOT fr} +%% +\LoadClassWithOptions{article} \ProcessOptions\relax +%% +\ifx\pdfoutput\undefined + \usepackage[latin1]{inputenc} + \usepackage[T1]{fontenc} + \usepackage{url} + \usepackage[dvips]{graphicx} + \newcommand{\phantomsection}{\null} %% provided by hyperref. +\else + \pdfoutput=1\pdfcompresslevel=9\pdfadjustspacing=1 + \usepackage[pdftex,bookmarks,a4paper,colorlinks]{hyperref} + \usepackage{aeguill} + \usepackage[pdftex]{graphicx} + \hypersetup{bookmarksnumbered,plainpages=false,hypertexnames=false} + \hypersetup{pagecolor=blue,linkcolor=blue,citecolor=blue,urlcolor=blue} + \hypersetup{pdfcreator=PDFLaTeX with pbsheet class} +\fi +\RequirePackage{amsmath,amsfonts,amssymb,amsthm,latexsym} +\RequirePackage[a4paper,portrait]{geometry} +\RequirePackage{rotating} +\RequirePackage{moreverb} +\RequirePackage[latin1]{inputenc} +\RequirePackage{xspace} +\RequirePackage[francais]{babel} +\RequirePackage{color} +\RequirePackage{listings} +\ifx\pdfoutput\undefined +\else %% listings/hyperref compatibility. + \makeatletter + \providecommand*{\toclevel@lstlisting}{0} + \makeatother +\fi +%%%% +%%%% Macros et Environments +%%%% +%% +\newcommand{\MFILE}[1]{ + \begin{lstinputlisting}[language=Matlab]{#1.m}% + \label{code:#1}% + \end{lstinputlisting} +} +%% +\newcommand{\SCEFILE}[1]{ + \begin{lstinputlisting}[language=Scilab]{#1.sce}% + \label{code:#1}% + \end{lstinputlisting} +} +%% +\newcommand{\SCIFILE}[1]{ + \begin{lstinputlisting}[language=Scilab]{#1.sci}% + \label{code:#1}% + \end{lstinputlisting} +} +%% +\newcommand{\FFIG}[5]{%% fname cm scale floatparams caption +%% here, floatparams is in the set {vflt,rflt,lflt}, cf. floatflt doc. + \begin{floatingfigure}[#4]{#2}% + \begin{center}% + \includegraphics[scale=#3]{#1}% + \caption{#5}% + \label{fi:#1}% + \end{center}% + \end{floatingfigure}% +} +%% +\newcommand{\FIG}[4]{%% fname scale floatparams caption + \begin{figure}[#3]% + \begin{center}% + \includegraphics[scale=#2]{#1}% + \caption{{#4}}% + \label{fi:#1}% + \end{center}% + \end{figure} +} +%% +\makeatletter% +\newcommand{\@TITRE}{Feuille de Travaux Pratiques} +\newcommand{\@AUTEUR}{Anonyme} +\newcommand{\@DATE}{\today} +\newcommand{\@FORMATION}{Agr\'egation de math\'ematiques% + -- Oral de mod\'elisation stochastique} +\newcommand{\@ANNEE}{Ann\'ee} +\newcommand{\@ETABLISSEMENT}{\'Etablissement} +\newcommand{\@WEB}{http://www.lsp.ups-tlse.fr/Chafai/agregteam.html} +\newcommand{\@MEL}{anonyme@pas.net} +\newcommand{\TITRE}[1]{% + \renewcommand{\@TITRE}{#1}} +\newcommand{\AUTEUR}[1]{% + \renewcommand{\@AUTEUR}{#1}} +\newcommand{\DATE}[1]{% + \renewcommand{\@DATE}{#1}} +\newcommand{\FORMATION}[1]{% + \renewcommand{\@FORMATION}{#1}} +\newcommand{\ANNEE}[1]{% + \renewcommand{\@ANNEE}{#1}} +\newcommand{\ETABLISSEMENT}[1]{% + \renewcommand{\@ETABLISSEMENT}{#1}} +\newcommand{\WEB}[1]{% + \renewcommand{\@WEB}{#1}} +\newcommand{\MEL}[1]{% + \renewcommand{\@MEL}{#1}} +\makeatother +%% +\makeatletter +\newcommand{\PREMIEREPAGE}{% + \newpage\null% + \begin{center}% + \hrule height .2pt% + \medskip{\Large\bfseries\@TITRE\par}\medskip% + \hrule height .2pt% + \end{center}% + \bigskip%\vfil% +} +\makeatother +%% +\theoremstyle{plain}% +\newtheorem{thm}{Theor\`eme}[section]% +\newtheorem{cor}[thm]{Corollaire}% +\newtheorem{prop}[thm]{Proposition}% +\newtheorem{lem}[thm]{Lemme}% +\theoremstyle{definition}% +\newtheorem{defi}[thm]{D\'efinition}% +\newtheorem{xpl}[thm]{Exemple}% +\newtheorem{exo}[thm]{Exercice}% +\newtheorem{hyp}[thm]{Hypoth\`ese}% +\newtheorem{eur}[thm]{Heuristique}% +\newtheorem{pro}[thm]{Probl\`eme}% +\theoremstyle{remark}% +\newtheorem{rem}[thm]{Remarque}% +%%%% +%%%% Commands to be executed before \begin{document} +%%%% +%% +\geometry{left=2.5cm,right=2.5cm} +%% +\lstset{extendedchars=true,% + basicstyle=\small,% + commentstyle=\ttfamily\color[rgb]{0,0,0.5},% + stringstyle=\color[rgb]{0,0.5,0},% + frame=tblr,frameround=tttt,% + %%labelstyle=\tiny,labelstep=1,labelsep=10pt +} +%% +\makeatletter +\renewcommand{\@evenfoot} +{\tiny\normalfont + \@DATE. Copyright ^^a9 \@AUTEUR\ \@MEL. GNU FDL Copyleft. +%% \url{http://www.gnu.org/copyleft/fdl.html}. + \hfil \upshape Page n^^b0{\thepage}.} +\renewcommand{\@evenhead} +{\tiny\scshape \@ETABLISSEMENT\ -- \@FORMATION\ -- + \normalfont\bfseries\@ANNEE. + \normalfont\hfil \upshape Page n^^b0{\thepage}.} +\renewcommand{\@oddfoot}{\@evenfoot} +\renewcommand{\@oddhead}{\@evenhead} +\makeatother +%% smaller sections titles +\makeatletter +\renewcommand{\section}{\@startsection% + {section}% % name + {1}% % level + {0em}% % indent + {\baselineskip}% % beforeskip + {0.5\baselineskip}% % afterskip + {\normalfont\large\bfseries}}% % style +\renewcommand{\subsection}{\@startsection% + {subsection}% % name + {2}% % level + {0em}% % indent + {\baselineskip}% % beforeskip + {0.25\baselineskip}% % afterskip + {\normalfont\bfseries}% % style +} +\makeatother +%% +%% Macros +%% +\newcommand{\DOIFPDF}[2]{\ifx\pdfoutput\undefined #2\else#1\fi} +%% For nice typesetting of the actual engine name. +\newcommand{\EngineName}{\DOIFPDF{\texttt{pdf}\LaTeX}{\LaTeXe}} +%% For short rough notes and warnings. +\newcommand{\NB}[1]{{\large\textbf{*** #1 ***}}} +%% To force maths mode. +\newcommand{\EM}{\ensuremath} +%%% Limits +\newcommand{\limLeb}[2] + {\underset{#1\to+\infty}{\overset{\bL^{#2}}{\longrightarrow}}} +\newcommand{\limE}[1] + {\underset{#1\to+\infty}{\overset{\text{^^e9tr.}}{\longrightarrow}}} +\newcommand{\limW}[1] + {\underset{#1\to+\infty}{\overset{\text{weak}}{\longrightarrow}}} +\newcommand{\limP}[1] + {\underset{#1\to+\infty}{\overset{\dP}{\longrightarrow}}} +\newcommand{\limL}[1] + {\underset{#1\to+\infty}{\overset{\cL}{\longrightarrow}}} +\newcommand{\limn}[1] + {\underset{#1\to+\infty}{\longrightarrow}} +\newcommand{\mylim}[2] + {\underset{#1\to#2}{\longrightarrow}} +\newcommand{\limPS}[1] + {\underset{#1\to+\infty}{\overset{\text{p.s.}}{\longrightarrow}}} +\newcommand{\limAS}[1] + {\underset{#1\to+\infty}{\overset{\text{a.s.}}{\longrightarrow}}} +%% Time +\makeatletter +\providecommand{\timenow}{\@tempcnta\time% +\@tempcntb\@tempcnta% +\divide\@tempcntb60% +\ifnum10>\@tempcntb0\fi\number\@tempcntb% +\multiply\@tempcntb60% +\advance\@tempcnta-\@tempcntb% +:\ifnum10>\@tempcnta0\fi\number\@tempcnta} +\makeatother +%% Mathmode double capital letters. +\newcommand{\dA}{\EM{\mathbb{A}}} +\newcommand{\dB}{\EM{\mathbb{B}}} +\newcommand{\dC}{\EM{\mathbb{C}}} +\newcommand{\dD}{\EM{\mathbb{D}}} +\newcommand{\dE}{\EM{\mathbb{E}}} +\newcommand{\dF}{\EM{\mathbb{F}}} +\newcommand{\dG}{\EM{\mathbb{G}}} +\newcommand{\dH}{\EM{\mathbb{H}}} +\newcommand{\dI}{\EM{\mathbb{I}}} +\newcommand{\dJ}{\EM{\mathbb{J}}} +\newcommand{\dK}{\EM{\mathbb{K}}} +\newcommand{\dL}{\EM{\mathbb{L}}} +\newcommand{\dM}{\EM{\mathbb{M}}} +\newcommand{\dN}{\EM{\mathbb{N}}} +\newcommand{\dO}{\EM{\mathbb{O}}} +\newcommand{\dP}{\EM{\mathbb{P}}} +\newcommand{\dQ}{\EM{\mathbb{Q}}} +\newcommand{\dR}{\EM{\mathbb{R}}} +\newcommand{\dS}{\EM{\mathbb{S}}} +\newcommand{\dT}{\EM{\mathbb{T}}} 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+\newcommand{\cV}{\EM{\mathcal{V}}} +\newcommand{\cW}{\EM{\mathcal{W}}} +\newcommand{\cX}{\EM{\mathcal{X}}} +\newcommand{\cY}{\EM{\mathcal{Y}}} +\newcommand{\cZ}{\EM{\mathcal{Z}}} +%% Mathmode bold capital letters. +\newcommand{\bA}{\EM{\mathbf{A}}} +\newcommand{\bB}{\EM{\mathbf{B}}} +\newcommand{\bC}{\EM{\mathbf{C}}} +\newcommand{\bD}{\EM{\mathbf{D}}} +\newcommand{\bE}{\EM{\mathbf{E}}} +\newcommand{\bF}{\EM{\mathbf{F}}} +\newcommand{\bG}{\EM{\mathbf{G}}} +\newcommand{\bH}{\EM{\mathbf{H}}} +\newcommand{\bI}{\EM{\mathbf{I}}} +\newcommand{\bJ}{\EM{\mathbf{J}}} +\newcommand{\bK}{\EM{\mathbf{K}}} +\newcommand{\bL}{\EM{\mathbf{L}}} +\newcommand{\bM}{\EM{\mathbf{M}}} +\newcommand{\bN}{\EM{\mathbf{N}}} +\newcommand{\bO}{\EM{\mathbf{O}}} +\newcommand{\bP}{\EM{\mathbf{P}}} +\newcommand{\bQ}{\EM{\mathbf{Q}}} +\newcommand{\bR}{\EM{\mathbf{R}}} +\newcommand{\bS}{\EM{\mathbf{S}}} +\newcommand{\bT}{\EM{\mathbf{T}}} +\newcommand{\bU}{\EM{\mathbf{U}}} +\newcommand{\bV}{\EM{\mathbf{V}}} +\newcommand{\bW}{\EM{\mathbf{W}}} +\newcommand{\bX}{\EM{\mathbf{X}}} +\newcommand{\bY}{\EM{\mathbf{Y}}} +\newcommand{\bZ}{\EM{\mathbf{Z}}} +%% Useful shortcuts to some Greek letters and symbols. +\newcommand{\al}{\alpha} +\newcommand{\be}{\beta} +\newcommand{\De}{\Delta} +\newcommand{\de}{\delta} +\newcommand{\ga}{\gamma} +\newcommand{\Ga}{\Gamma} +\newcommand{\g}{\gamma} +\newcommand{\gn}{\g_n} +\newcommand{\gt}[1]{\g^{\otimes #1}} +\newcommand{\la}{\lambda} +\newcommand{\La}{\Lambda} +\newcommand{\lan}{\la_n} +\newcommand{\na}{\nabla} +\newcommand{\Om}{\Omega} +\newcommand{\om}{\omega} +\newcommand{\ph}{\Phi} +\newcommand{\Si}{\Sigma} +\newcommand{\si}{\sigma} +\newcommand{\Te}{\Theta} +\newcommand{\te}{\theta} +\newcommand{\ta}{\tau} +\newcommand{\Ups}{\Upsilon} +\newcommand{\veps}{\varepsilon} +\newcommand{\vphi}{\varphi} +\newcommand{\bul}{\EM{\bullet}} +%% Functional prototype with dimensioning. +\newcommand{\p}[4]{{#3}\!\left#1{#4}\right#2} +%% Norms and related. +\newcommand{\ABS}[1]{\EM{{\left| #1 \right|}}} % |1| +\newcommand{\BRA}[1]{\EM{{\left\{#1\right\}}}} % {1} +\newcommand{\DP}[1]{\EM{{\left<#1\right>}}} % <1> +\newcommand{\NRM}[1]{\EM{{\left\| #1\right\|}}} % ||1|| +\newcommand{\NI}[1]{\EM{{\left\| #1\right\|}_\infty}} % infinite norm. +\newcommand{\OSC}[1]{\EM{{\p(){\mathrm{osc}}{#1}}}} % oscillation. +\newcommand{\PAR}[1]{\EM{{\left(#1\right)}}} % (1) +\newcommand{\BPAR}[1]{\EM{{\biggl(#1\biggr)}}} % (1) +\newcommand{\BABS}[1]{\EM{{\biggl|#1\biggr|}}} % (1) +\newcommand{\pd}{\EM{\partial}} % derivee partielle +\newcommand{\PD}[2]{\EM{{\frac{\partial #1}{\partial #2}}}}%part. deriv. +\newcommand{\SBRA}[1]{\EM{{\left[#1\right]}}} % [1] +\newcommand{\VT}[1]{\EM{{\|#1\|}_{\text{\scriptsize VT}}}} % tot. var. +\newcommand{\LIP}[1]{\EM{\|#1\|_{\mathrm{Lip}}}} % Lipschitz norm. +\newcommand{\WH}[1]{\widehat{#1}} +%% Fonctionals and functions. +\newcommand{\sentf}{\bH} +\newcommand{\sent}[1]{\p(){\sentf}{#1}} +\newcommand{\bentf}[1]{\bH_{#1}} +\newcommand{\bent}[2]{\p(){\bentf{#1}}{#2}} +\newcommand{\eentf}{\bN} +\newcommand{\eent}[1]{\p(){\eentf}{#1}} +\newcommand{\entf}[1]{\mathbf{Ent}_{#1}} +\newcommand{\ent}[2]{\p(){\entf{#1}}{#2}} +\newcommand{\ientf}[1]{\bH^{#1}} +\newcommand{\ient}[2]{\p(){\ientf{#1}}{#2}} +\newcommand{\rentf}{\mathbf{Ent}} +\newcommand{\rent}[2]{\p(){\rentf}{#1\,\vert\,#2}} +\newcommand{\entr}{\mathbf{Ent}_r} +\newcommand{\enef}[1]{\boldsymbol{\mathcal{E}}_{#1}} +\newcommand{\ene}[2]{\p(){\enef{#1}}{#2}} +\newcommand{\imutf}{\bI} +\newcommand{\imut}[1]{\p(){\imutf}{#1}} +\newcommand{\fcrf}{\dI} +\newcommand{\fcr}[1]{\p(){\fcrf}{#1}} +\newcommand{\fishf}{\bJ} +\newcommand{\fish}[1]{\p(){\fishf}{#1}} +\newcommand{\fishmf}{\dJ} +\newcommand{\fishm}[1]{\p(){\fishmf}{#1}} +\newcommand{\moyf}[1]{\bE_{#1}} +\newcommand{\moy}[2]{\p(){\moyf{#1}}{#2}} +\newcommand{\econd}[2]{\p(){\moyf{}}{#1\,\vert\,#2}} +\newcommand{\bmoy}[2]{\moyf{#1}\!\biggl(#2\biggr)} +\newcommand{\corrf}[1]{\mathbf{Cor}_{#1}} +\newcommand{\corr}[3]{\p(){\corrf{#1}}{#2,#3}} +\newcommand{\covf}[1]{\mathbf{Cov}_{#1}} +\newcommand{\cov}[3]{\p(){\covf{#1}}{#2,#3}} +\newcommand{\bcov}[3]{\covf{#1}\!\biggl(#2,#3\biggr)} +\newcommand{\suppf}{\mathrm{supp}} +\newcommand{\supp}[1]{\suppf\PAR{#1}} +\newcommand{\DIMF}{\mathrm{dim}} +\newcommand{\DIM}[1]{\DIMF\PAR{#1}} +\newcommand{\RANGEF}{\mathrm{range}} +\newcommand{\RANGE}[1]{\RANGEF\PAR{#1}} +\newcommand{\varf}[1]{\mathbf{Var}_{#1}} +\newcommand{\var}[2]{\p(){\varf{#1}}{#2}} +\newcommand{\bvar}[2]{\varf{#1}\!\biggl(#2\biggr)} +\newcommand{\Kf}{\bK} +\newcommand{\K}[1]{\p(){\Kf}{#1}} +\newcommand{\vrs}[1]{\mathbf{L}_{#1}} +\newcommand{\lik}[1]{\vrs{#1}} +%% Sets, spaces, and functional spaces. +\newcommand{\C}[1]{\p(){\cC}{#1}} +\newcommand{\Cb}[1]{\p(){\cC_b}{#1}} +\newcommand{\Cc}[1]{\p(){\cC_c}{#1}} +\newcommand{\Cn}[2]{\p(){\cC^{#1}}{#2}} +\newcommand{\Ci}[1]{\Cn{\infty}{#1}} +\newcommand{\Cic}[1]{\p(){\cC_c^\infty}{#1}} +\newcommand{\Cnc}[2]{\p(){\cC_c^{#1}}{#2}} +\newcommand{\Cnb}[2]{\p(){\cC_b^{#1}}{#2}} +\newcommand{\Cib}[1]{\p(){\cC_b^\infty}{#1}} +\newcommand{\leb}[2]{\p(){\bL^{#1}}{#2}} +\newcommand{\lebb}[1]{\bL^{#1}} +%% Determinant, trace, ... +\newcommand{\Tr}{\mathrm{Tr\,}} +\newcommand{\Det}[1]{\mathrm{Det}\,} +\newcommand{\TR}[1]{\p(){\mathrm{Tr}}{#1}} +\newcommand{\DIAG}[1]{\p(){\mathrm{Diag}}{#1}} +\newcommand{\DET}[1]{\p(){\mathrm{Det}}{#1}} +\newcommand{\SIG}[1]{\p(){\mathrm{Sign}}{#1}} +\newcommand{\KER}[1]{\p(){\mathrm{Ker}}{#1}} +\newcommand{\RG}[1]{\p(){\mathrm{Range}}{#1}} +\newcommand{\DIAM}[1]{\p(){\mathrm{Diam}}{#1}} +\newcommand{\ID}{\mathbf{Id}} +\newcommand{\Id}{\mathrm{Id}} +\newcommand{\Mo}{\mathbf{1}} +\newcommand{\Vo}{\mathrm{1}} +%%% Semi-groups, generators, carre-du-champ, ... +\newcommand{\CD}{CD(\rho ,\infty)} +\newcommand{\GA}{\boldsymbol{\Gamma}} +\newcommand{\GD}{\GA_{\!\!{\mathbf 2}}} +\newcommand{\GI}{\bL} +\newcommand{\GR}{\nabla} +\newcommand{\GIV}{\EM{\overrightarrow{\GI}}} +\newcommand{\GIB}{\EM{\overline{\GI}}} +\newcommand{\LA}{\boldsymbol{\Delta}} +\newcommand{\ROT}{\mathbf{rot}\,} +\newcommand{\DIV}{\mathbf{div}\,} +\newcommand{\PT}[1][t]{\mathbf{P}_{\!#1}} +\newcommand{\SGf}[1]{{\mathbf P}_{#1}} +\newcommand{\SG}[2]{\p(){\SGf{\!#1}}{#2}} +\newcommand{\SGQf}[1]{{\mathbf Q}_{#1}} +\newcommand{\SGQ}[2]{\p(){\SGQf{#1}}{#2}} +\newcommand{\isopf}{\mathbf{I}} +\newcommand{\isop}[1]{\p(){\isopf}{#1}} +\newcommand{\disop}[1]{\p(){\isopf'}{#1}} +\newcommand{\ddisop}[1]{\p(){\isopf''}{#1}} +\newcommand{\HESS}[1]{\GR^2\!{#1}} +\newcommand{\Hess}[1]{{\p(){\mathrm{Hess}}{#1}}} +\newcommand{\HAM}{\bH} +\newcommand{\POT}{\bV} +\newcommand{\PF}{\bZ} +\newcommand{\DOM}{\EM{\cD_2(\GI)}} +\newcommand{\laV}{\EM{\overrightarrow{\la}}} +\newcommand{\DOML}{\EM{\cD_2\PAR{\GI}}} +\newcommand{\DOMB}{\EM{\cD_2(\GIB)}} +\newcommand{\DOMV}{\EM{\cD_2(\GIV)}} +\newcommand{\GAV}{\EM{\overrightarrow{\GA}}} +\newcommand{\D}{\mathbf{D}} +%% Topology. +\newcommand{\inter}[1]{{\overset{\circ}{#1}}} % interior +\newcommand{\adher}[1]{{\overline{#1}}} % adherence +\newcommand{\ADH}[1]{\mathbf{adh}(#1)} +\newcommand{\INT}[1]{\mathbf{int}(#1)} +%% Miscellany. +\newcommand{\DOTIMES}{\otimes\cdots\otimes} +\newcommand{\DOPLUS}{\oplus\cdots\oplus} +\newcommand{\CARD}{\mathrm{card}} +\newcommand{\SSK}[1]{\substack{#1}} +\renewcommand{\leq}{\leqslant} +\renewcommand{\geq}{\geqslant} +\newcommand{\bs}{\EM{\backslash}} +\newcommand{\Inf}{\boldsymbol{\inf}} +\newcommand{\Sup}{\boldsymbol{\sup}} +\newcommand{\ds}[1]{\EM{\displaystyle{#1}}} +\newcommand{\eg}{\overset{\Delta}{=}} +\newcommand{\fdefeq}{\overset{\text{d^^e9f.}}{=}} +\newcommand{\edefeq}{\overset{\text{def.}}{=}} +\newcommand{\defeq}{:=} +\newcommand{\equ}{\; \Leftrightarrow \;} +\newcommand{\ex}{\exists \,} +\newcommand{\imp}{\Rightarrow \;} +\newcommand{\ssi}{{\it ssi}} +\newcommand{\tout}{\forall \,} +\newcommand{\tq}{\,|\,} +\newcommand{\st}{\,|\,} +\newcommand{\DSFRAC}[2]{\EM{\frac{\ds{#1}}{\ds{#2}}}} +\newcommand{\SR}[2]{\EM{\strackrel{#1}{#2}}} +\def\Ieme{\up{\lowercase{i^^e8me}}\xspace} +%% Vectors and matrices. +\newcommand{\VECTOR}[1]{\PAR{\begin{array}{c} #1 \end{array}}} +\newcommand{\BIMATRIX}[1]{\PAR{\begin{array}{cc} #1 \end{array}}} +\newcommand{\Vector}[2]{\PAR{\begin{array}{#1} #2 \end{array}}} +\newcommand{\Bimatrix}[2]{\PAR{\begin{array}{#1} #2 \end{array}}} +%%%% +%%%% Commands to be executed just after \begin{document} +%%%% +\AtBeginDocument{% + \selectlanguage{francais}% + \graphicspath{{images/}{img/}{Images}{Img}{ps}{eps}{PS}{EPS}% + {postscript}{postscripts}{pdf}{PDF}}% + \PREMIEREPAGE% +} +%%%% +%%%% Commands to be executed just before \end{document} +%%%% +\AtEndDocument{\vfill} +\endinput +%% +%% End of file `pbsheet.cls'. diff --git a/macros/latex/contrib/pbsheet/xpl/pgm/probadis.m b/macros/latex/contrib/pbsheet/xpl/pgm/probadis.m new file mode 100644 index 0000000000..7f472ad404 --- /dev/null +++ b/macros/latex/contrib/pbsheet/xpl/pgm/probadis.m @@ -0,0 +1,65 @@ +function res = probadis(x,p) +%res = PROBADIS(x,p) +% Input x vector of real numbers (support points in IR) +% p vector of probability weights associated to x +% i.e. non-negative real numbers such that sum(p) == 1 +% Output res random number chosen from the finite discrete +% distribution on x(1),...,x(n) with probability weights +% p(1),...,p(n) where n == length(x) == length(p) +% +% Renvoie UNE réalisation de la loi discrète à support fini sur IR dont les +% points de support sont les composantes du vecteur x, et les poids sont les +% composantes du vecteur p. Donc x(i) a une probabilité p(i) d'être renvoyé. +% +% La méthode consiste à écrire l'intervalle [0,1] comme une réunion disjointe +% d'intervalles de longueurs p(1),..., p(n) puis à regarder au quel appartient +% la réalisation d'une loi uniforme obtenue par un appel à rand. Les valeurs +% renvoyées par des appels successifs sont donc pseudo-indépendantes. +% Pour des raisons d'efficacité, les conditions nécessaires suivantes ne sont +% pas contrôlées par cette fonction : +% - le nombre de paramètres passés est exactement 2 +% - les deux paramètres x et p sont bien des vecteurs et sont de même longueur +% - les composantes de p sont positives ou nulles et leur somme vaut 1. +% +% See also RDISCR. +% +% ### Copyright (C) D. Chafaï, 2003-12-06. +% ### http://www.lsp.ups-tlse.fr/Chafai/agregation.html +% ### Licence GNU General Public License http://www.gnu.org/copyleft/gpl.html +% + +% On pourrait implémenter cette fonction de la façon suivante : +% +% INDICES = find(cumsum(p) >= rand); +% res = x(INDICES(1)); +% return; +% +% Cette méthode est correcte mais inefficace car elle ne tient pas compte de +% la monotonie de cumsum(p), ce qui entraine des tests inutiles une fois que +% la valeur critique a été franchie. En jargon, c'est un 'firstmatch' qu'il +% nous faut, pas un 'matchall'. Le nombre de if impliqués dans find est +% toujours égal à la taille de ce qu'on lui passe en paramètre. +% +% Morale de l'histoire : la brièveté d'un code n'assure pas sa performance ! +% Et l'abscence de if dans un code ne signifie pas qu'il ne fait pas appel +% indirectement à des if, et encore moins que cela est fait de manière +% optimale ! La fonction find n'est qu'une boucle for contenant un if. +% +% Ci-dessous, nous utilisons une version plus rapide dans le plus pur style +% for-if avec un nombre de if optimal. On pourrait adapter l'ordre des +% intervalles testés (et donc l'arbre associéé) aux poids p(i) de façon à +% tester d'abord les intervalles les plus probables. Est-ce vraiment mieux ? +% Exercice ! + +n = length(x); % le nombre d'atomes. +r = rand; % une réalisation de loi uniforme sur [0,1]. +a = 0; b = p(1); % [a,b] = sous-intervalle de proba p(i) pour l'uniforme. +for i = 1:n-1 % parcours de tous les sous-intervalles juxtaposés. + if ((r >= a) & (r < b)) + res = x(i); + return; % on a trouvé le bon intervalle, on sort. + end + a = b; b = b + p(i+1); % on passe à l'intervalle suivant. +end +res = x(n); % le bon intervalle est le dernier. +return; diff --git a/macros/latex/contrib/pbsheet/xpl/pgm/rdiscr.m b/macros/latex/contrib/pbsheet/xpl/pgm/rdiscr.m new file mode 100644 index 0000000000..b3346178e4 --- /dev/null +++ b/macros/latex/contrib/pbsheet/xpl/pgm/rdiscr.m @@ -0,0 +1,72 @@ +function X = rdiscr(num,x,p) +%X = RDISCR(num,x,p) +% Input num positive integer or a vector [lig,col] of integers +% x vector of real numbers (support points in IR) +% p vector of probability weights associated to x +% i.e. non-negative real numbers such that sum(p) == 1 +% Output X num-vector or a num-matrix of random numbers +% chosen from the finite discrete distribution on +% x(1),...,x(n) with probability weights p(1),...,p(n) +% where n == length(x) == length(p) +% +% Renvoie num réalisations ou une matrice [lig,col] de réalisations +% pseudo-i.i.d de la loi discrète à support fini sur IR dont les +% points de support sont les composantes du vecteur x, et les poids sont les +% composantes du vecteur p. Donc x(i) a une probabilité p(i) d'être renvoyé. +% +% La méthode consiste à écrire l'intervalle [0,1] comme une réunion disjointe +% d'intervalles de longueurs p(1),..., p(n) puis à regarder au quel appartient +% la réalisation d'une loi uniforme obtenue par un appel à rand. Les valeurs +% renvoyées par des appels successifs sont donc pseudo-indépendantes. +% Pour des raisons d'efficacité, les conditions nécessaires suivantes ne sont +% pas contrôlées par cette fonction : +% - le nombre de paramètres passés est exactement 3 +% - les deux paramètres x et p sont bien des vecteurs et sont de même longueur +% - les composantes de p sont positives ou nulles et leur somme vaut 1 +% - le paramètre num est un entier positif non nul une un couple de ce type +% +% See also PROBADIS. +% +% ### Copyright (C) D. Chafaï, 2003-12-06. +% ### http://www.lsp.ups-tlse.fr/Chafai/agregation.html +% ### Licence GNU General Public License http://www.gnu.org/copyleft/gpl.html +% + +% Voir les commentaires dans le code de la fonction probadis. +% Faire 'type probadis' pour cela. +% Le code qui suit pourrait beaucoup gagner en rapidité sur une machine // +% Il est possible de l'améliorer en stockant un arbre construit avec les +% tests utilisés par les valeurs déjà générées. Il est surtout aussi possible +% d'imiter le code matriciel de la fonction de répartition inverse binomiale +% de la fonction qbinom de Stixbox, appelée par rbinom. Exercice ! +% +% D'autres algorithmes sont possibles. Par exemple, on pourrait procéder +% avec des réalisations de Bernoulli i.i.d. (obtenues facilement avec rand) +% pour choisir l'intervalle : +% La probabilité d'être <= x(1) est p(1) +% Sinon, la probabilité d'être <= x(2) est p(2)/sum(p(2:n)) +% etc. Nul besoin de commencer par x(1), et l'on peut adapter l'arbre utilisé +% aux poids p(i) de façon à faire un nombre de tests optimal. Exercice ! + +if length(num) == 1 + num = [num 1]; +else + num = reshape(num,1,2); +end + +n = length(x); % le nombre d'atomes. +U = rand(num); % réalisations de loi uniforme sur [0,1]. +X = repmat(x(n),num(1),num(2)); % par défaut, la valeur est la plus grande +for l = 1:num(1) % lignes +for c = 1:num(2) % colonnes + a = 0; b = p(1); % [a,b] = sous-inter. de proba p(i) pour l'uniforme. + for i = 1:n-1 % parcours de tous les sous-intervalles juxtaposés. + if ((U(l,c) >= a) & (U(l,c) < b)) + X(l,c) = x(i); + break; % on a trouvé le bon intervalle, on sort. + end + a = b; b = b + p(i+1); % on passe à l'intervalle suivant. + end +end +end +return; diff --git a/macros/latex/contrib/pbsheet/xpl/pgm/rint.m b/macros/latex/contrib/pbsheet/xpl/pgm/rint.m new file mode 100644 index 0000000000..9e8dd3c850 --- /dev/null +++ b/macros/latex/contrib/pbsheet/xpl/pgm/rint.m @@ -0,0 +1,25 @@ +function X = rint(num,k); +% X = RINT(num,k) +% Input num positive integer or a vector [lig,col] of integers +% k positive integer +% Output X num-vector or a num-matrix of random numbers +% chosen from uniform distribution on {1,...,k} +% +% Renvoie num réalisations ou une matrice [lig,col] de réalisations +% pseudo-indépendantes de la loi uniforme discrète sur les k premiers +% entiers non nuls {1,...,k}. +% +% La méthode consiste à considérer la partie entière de 1+kU où U suit une loi +% uniforme sur [0,1]. Cette dernière s'obtient via la fonction rand, et les +% valeurs renvoyées par des appels successifs sont donc pseudo-indépendantes. +% +% ### Copyright (C) D. Chafaï, 2003-12-06. +% ### http://www.lsp.ups-tlse.fr/Chafai/agregation.html +% ### Licence GNU General Public License http://www.gnu.org/copyleft/gpl.html +% + +if length(num) == 1 + num = [num 1]; +end +X = ceil(k * rand(num)); +return; diff --git a/macros/latex/contrib/pbsheet/xpl/pgm/simbin.m b/macros/latex/contrib/pbsheet/xpl/pgm/simbin.m new file mode 100644 index 0000000000..f9c618a8a6 --- /dev/null +++ b/macros/latex/contrib/pbsheet/xpl/pgm/simbin.m @@ -0,0 +1,55 @@ +% +% ### Copyright (C) D. Chafaï, 2003-12-06. +% ### http://www.lsp.ups-tlse.fr/Chafai/agregation.html +% ### Licence GNU General Public License http://www.gnu.org/copyleft/gpl.html +% +% Ce bout de code permet de simuler une loi binomiale B(n,p) et de comparer +% graphiquement les moyennes empiriques avec la moyenne théorique. +% Tous les vecteurs sont des vecteurs ligne. +% +% Nota bene : la bibliothèque Stixbox fournit la fonction rbinom qui permet +% de s'affranchir des calculs détaillés ici, cf. fin du présent fichier. +% +clear r n p q P Q C B X +clf +% +r = input('Nombre maximal de réalisations ? = '); +n = input('Taille n de la loi binomiale B(n,p), qui a n+1 atomes ? = '); +p = input('Valeur du paramètre p de la loi binomiale B(n,p) ? = '); +% p = proba de gagner à pile ou face = proba de 1 dans la Bernoulli sur {0,1} +% q = proba de perdre à pile ou face = proba de 0 dans la Bernoulli sur {0,1} +% B(n,p) est la loi de la somme de n v.a. i.i.d. de Bernoulli de ce type, +% i.e. la nième puissance de convolution de cette loi de Bernoulli. Elle +% représente la loi du nombre de succés à pile ou face en n lancés. +% +disp(sprintf('Calcul de la loi binomiale B(%d,%f)',n,p)) +q = 1 - p; +P = [1 , cumprod(p * ones(1,n))]; % puissances croissantes de p +Q = [fliplr(cumprod(q * ones(1,n))), 1]; % puissances décroissantes de q +C = [1 , cumprod([n:-1:1]) ./ cumprod([1:n])]; % coef. du binôme de i = 1 à n +B = C .* P .* Q; % vecteur des poids de B(n,p) +% +disp(sprintf('Génération aléatoire de %d réalisations de B(%d,%f)',r,n,p)) +X = rdiscr([1,r],[0:n],B); % échantillonnage +% alternative : +%X=[] +%for i=1:r +% X = [X probadis([0:n],B)]; +%end +% +disp(sprintf('Tracé des graphique.')) +plot(cumsum(X) ./ [1:length(X)],'b') % tracé des moyennes empiriques +title('Loi des Grands Nombres') +xlabel('Nombre de realisations') +ylabel('Moyennes empiriques') +hold on +plot(n * p * ones(1,r),'r--') % tracé de la moyenne théorique +legend('Empirique','Theorique') +hold off +% +% Avec Stixbox, inutile de calculer B et d'appeler rdiscr([1,r],[0:n],B) +% puisque qu'un simple rbinom([1,r],n,p) suffit. +% Même si l'on décide d'utiliser quand même prodadis, les coefficients +% binomiaux nécessaires au calcul de B peuvent se calculer beaucoup plus vite +% en utilisant la fonction Stixbox bincoef qui fait appel à la fonction gamma. +% diff --git a/macros/latex/contrib/pbsheet/xpl/xpl-fr.bib b/macros/latex/contrib/pbsheet/xpl/xpl-fr.bib new file mode 100644 index 0000000000..076f05db04 --- /dev/null +++ b/macros/latex/contrib/pbsheet/xpl/xpl-fr.bib @@ -0,0 +1,39 @@ +@book{barbe-ledoux, + AUTHOR = {Barbe, Ph. and Ledoux, M.}, + TITLE = {Probabilit{\'e}s}, + YEAR = {1998}, + PAGES = {262}, + PUBLISHER = {Belin}, + SERIES = {De la licence {\`a} l'agr{\'e}gation}, + ISBN = {2-7011-2131-0} +} + +@book{dacunha-castelle-duflo, + AUTHOR = {Dacunha-Castelle, D. and Duflo, M.}, + TITLE = {Probabilit{\'e}s et statistiques. {T}ome 1}, + NOTE = {Probl{\`e}mes {\`a} temps fixe.}, + PUBLISHER = {Masson}, + ADDRESS = {Paris}, + YEAR = {1982}, + PAGES = {xii+214}, + ISBN = {2-225-74547-1}, +} + +@Book{bouleau, + AUTHOR = {Bouleau, N.}, + TITLE = {Probabilit{\'e}s de l'ing{\'e}nieur, + variables al{\'e}atoires et simulation}, + PUBLISHER = {Hermann}, + YEAR = {1986} +} + +@Incollection{ycart, + AUTHOR = {Ycart, B.}, + TITLE = {{M}od\`eles et {A}lgorithmes {M}arkoviens}, + PUBLISHER = {Springer}, + EDITOR = {{S}oci\'et\'e de {M}ath\'ematiques + {A}ppliqu\'ee et {I}ndustrielles}, + YEAR = {2002}, + SERIES = {Math\'ematiques et {A}pplications}, + VOLUME = {39} +} diff --git a/macros/latex/contrib/pbsheet/xpl/xpl-fr.dvi b/macros/latex/contrib/pbsheet/xpl/xpl-fr.dvi Binary files differnew file mode 100644 index 0000000000..bb9b08a0ad --- /dev/null +++ b/macros/latex/contrib/pbsheet/xpl/xpl-fr.dvi diff --git a/macros/latex/contrib/pbsheet/xpl/xpl-fr.pdf b/macros/latex/contrib/pbsheet/xpl/xpl-fr.pdf Binary files differnew file mode 100644 index 0000000000..515b993882 --- /dev/null +++ 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All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.00B) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMR5) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle 0 def +/isFixedPitch false def +end readonly def +/FontName /CMR5 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 0 /.notdef put +readonly def +/FontBBox{-341 -250 1304 965}readonly def +/UniqueID 5000788 def +currentdict end +currentfile eexec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cleartomark +%%EndFont +%%BeginFont: MSBM10 +%!PS-AdobeFont-1.1: MSBM10 2.1 +%%CreationDate: 1993 Sep 17 11:10:37 +% Math Symbol fonts were designed by the American Mathematical Society. +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (2.1) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (MSBM10) readonly def +/FamilyName (Euler) readonly def +/Weight (Medium) readonly def +/ItalicAngle 0 def +/isFixedPitch false def +end readonly def +/FontName /MSBM10 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 78 /N put +dup 80 /P put +readonly def +/FontBBox{-55 -420 2343 920}readonly def +/UniqueID 5031982 def +currentdict end +currentfile eexec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cleartomark +%%EndFont +%%BeginFont: MSAM7 +%!PS-AdobeFont-1.1: MSAM7 2.1 +%%CreationDate: 1992 Oct 17 08:30:29 +% Math Symbol fonts were designed by the American Mathematical Society. +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (2.1) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (MSAM7) readonly def +/FamilyName (Euler) readonly def +/Weight (Medium) readonly def +/ItalicAngle 0 def +/isFixedPitch false def +end readonly def +/FontName /MSAM7 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 54 /lessorequalslant put +dup 62 /greaterorequalslant put +readonly def +/FontBBox{0 -576 1569 1076}readonly def +/UniqueID 5032012 def +currentdict end +currentfile eexec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cleartomark +%%EndFont +%%BeginFont: CMMI5 +%!PS-AdobeFont-1.1: CMMI5 1.100 +%%CreationDate: 1996 Aug 02 08:21:10 +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.100) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMMI5) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle -14.04 def +/isFixedPitch false def +end readonly def +/FontName /CMMI5 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 0 /.notdef put +readonly def +/FontBBox{37 -250 1349 750}readonly def +/UniqueID 5087380 def +currentdict end +currentfile eexec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cleartomark +%%EndFont +%%BeginFont: CMR7 +%!PS-AdobeFont-1.1: CMR7 1.0 +%%CreationDate: 1991 Aug 20 16:39:21 +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.0) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMR7) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle 0 def +/isFixedPitch false def +end readonly def +/FontName /CMR7 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 0 /.notdef put +readonly def +/FontBBox{-27 -250 1122 750}readonly def +/UniqueID 5000790 def +currentdict end +currentfile eexec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cleartomark +%%EndFont +%%BeginFont: CMEX10 +%!PS-AdobeFont-1.1: CMEX10 1.00 +%%CreationDate: 1992 Jul 23 21:22:48 +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.00) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMEX10) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle 0 def +/isFixedPitch false def +end readonly def +/FontName /CMEX10 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 80 /summationtext put +readonly def +/FontBBox{-24 -2960 1454 772}readonly def +/UniqueID 5000774 def +currentdict end +currentfile eexec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cleartomark +%%EndFont +%%BeginFont: MSBM7 +%!PS-AdobeFont-1.1: MSBM7 2.1 +%%CreationDate: 1992 Oct 17 08:30:50 +% Math Symbol fonts were designed by the American Mathematical Society. +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (2.1) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (MSBM7) readonly def +/FamilyName (Euler) readonly def +/Weight (Medium) readonly def +/ItalicAngle 0 def +/isFixedPitch false def +end readonly def +/FontName /MSBM7 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 78 /N put +readonly def +/FontBBox{0 -504 2615 1004}readonly def +/UniqueID 5032014 def +currentdict end +currentfile eexec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cleartomark +%%EndFont +%%BeginFont: CMSY7 +%!PS-AdobeFont-1.1: CMSY7 1.0 +%%CreationDate: 1991 Aug 15 07:21:52 +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.0) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMSY7) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle -14.035 def +/isFixedPitch false def +end readonly def +/FontName /CMSY7 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 0 /.notdef put +readonly def +/FontBBox{-15 -951 1252 782}readonly def +/UniqueID 5000817 def +currentdict end +currentfile eexec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cleartomark +%%EndFont +%%BeginFont: CMMI7 +%!PS-AdobeFont-1.1: CMMI7 1.100 +%%CreationDate: 1996 Jul 23 07:53:53 +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.100) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMMI7) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle -14.04 def +/isFixedPitch false def +end readonly def +/FontName /CMMI7 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 0 /.notdef put +readonly def +/FontBBox{0 -250 1171 750}readonly def +/UniqueID 5087382 def +currentdict end +currentfile eexec +D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE +3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B +532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 +B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B +986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE +D919C2DDD26BDC0D99398B9F4D03D77639DF1232A4D6233A9CAF69B151DFD33F +C0962EAC6E3EBFB8AD256A3C654EAAF9A50C51BC6FA90B61B60401C235AFAB7B 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All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (2.1) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.0) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMSY9) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle -14.035 def +/isFixedPitch false def +end readonly def +/FontName /CMSY9 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 0 /.notdef put +readonly def +/FontBBox{-30 -958 1146 777}readonly def +/UniqueID 5000819 def +currentdict end +currentfile eexec +D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 +7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 +A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 +E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A +221A37D9A807DD01161779DDE7D31FF2B87F97C73D63EECDDA4C49501773468A +27D1663E0B62F461F6E40A5D6676D0037D33F24E2FAC2B0009AD3C8350CDF8CC +65BCA87979C36D14CB552E9A985E48BE4E88ECA16DF418749AF04FDD2B0E1380 +D281BB2476BB45FF30946B247DFD7F57305FA87E50CA338121C71CDFDF927A9C +77FF14CB4A1D6D80356FB1171ED38C37702350497B44E42CE31DB2F493807DAA +15B887C671199A54C4C1294BC520F5538C15556BC43C9F62342B121C6DCD6C5F +491DA47FF360201EE21C08A781ED0589A6DF91B99FE118B9B29E4F068672E52F +1A06C514D91C4C937D4E642503392B1CD1BF5AF0BCA28EBD840AD76CC39AD7AA +CF2C057D436A245217829DB786A3D592C0F5A073E44D137405B0D44F7D7356F1 +DFE3D933072E0CA836B5B72B1DDEB8FF86D5F9A2AFDEC789B2FBDAC307F2DBF7 +2B5BE20870A423F85DAC7075769E74EC93D759E9095E4C1FA2D5D8FCE0EB3586 +E193EE511DC1B950972544335EA5B46B3AF5B3FA5B7922DDAEFC5DB52B22CBCA +7C8ACA790FBE3B57A89D10E84DA86BCA85AD169F0B7F8F93B228EF212069866B +6D930B7FE660F87BB45A1DFF541F6B8926AFC7FA90E81B7628D4273B10474731 +4F5D42B1C5EEE258EF4C5255BAE05301CF5DA133FDD927D3065ADC77D0187C41 +7F0DC217781C1D37053C2D43C838597E76AB194348E78C59ED1AF4300B3885C2 +3408811AEC8C98E7E7E2E12CECEA0E666C993A48BE5CA6A23524B15433779571 +56CADB29D1963EB1AFE4EDDD12CC64264A6D12E624DC132B00CD3565784EC293 +83B0A827CD6D16BDDE351C14092435A687A98FC4599C6B712CD5809D8A8FB171 +BB2694D50BF5C52E068A58F907F702769A98ED6AA915DA9E56620FC819AF54A0 +30C2B7541C94336DC23F290F8A3C4B37E2C4513CACF349 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +cleartomark +%%EndFont +%%BeginFont: LCIRCLE10 +%!PS-AdobeFont-1.1: LCIRCLE10 1.001 +%%CreationDate: 1992 Oct 23 20:21:59 +%%RevisionDate: 2001 Jun 05 20:21:59 +% Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.001) readonly def +/Notice (Copyright (C) 1997, 2001 American Mathematical Society. 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All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.0) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. 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+gnudict begin +/Color true def +/Solid true def +/gnulinewidth 5.000 def +/userlinewidth gnulinewidth def +/vshift -46 def +/dl {10 mul} def +/hpt_ 31.5 def +/vpt_ 31.5 def +/hpt hpt_ def +/vpt vpt_ def +/M {moveto} bind def +/L {lineto} bind def +/R {rmoveto} bind def +/V {rlineto} bind def +/vpt2 vpt 2 mul def +/hpt2 hpt 2 mul def +/Lshow { currentpoint stroke M + 0 vshift R show } def +/Rshow { currentpoint stroke M + dup stringwidth pop neg vshift R show } def +/Cshow { currentpoint stroke M + dup stringwidth pop -2 div vshift R show } def +/UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def + /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def +/DL { Color {setrgbcolor Solid {pop []} if 0 setdash } + {pop pop pop Solid {pop []} if 0 setdash} ifelse } def +/BL { stroke userlinewidth 2 mul setlinewidth } def +/AL { stroke userlinewidth 2 div setlinewidth } def +/UL { dup gnulinewidth mul /userlinewidth exch def + dup 1 lt {pop 1} if 10 mul /udl exch def } def +/PL { stroke userlinewidth setlinewidth } def +/LTb { BL [] 0 0 0 DL } def +/LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def +/LT0 { PL [] 1 0 0 DL } def +/LT1 { PL [4 dl 2 dl] 0 1 0 DL } def +/LT2 { PL [2 dl 3 dl] 0 0 1 DL } def +/LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def +/LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def +/LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def +/LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def +/LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def +/LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def +/Pnt { stroke [] 0 setdash + gsave 1 setlinecap M 0 0 V stroke grestore } def +/Dia { stroke [] 0 setdash 2 copy vpt add M + hpt neg vpt neg V hpt vpt neg V + hpt vpt V hpt neg vpt V closepath stroke + Pnt } def +/Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V + currentpoint stroke M + hpt neg vpt neg R hpt2 0 V stroke + } def +/Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M + 0 vpt2 neg V hpt2 0 V 0 vpt2 V + hpt2 neg 0 V closepath stroke + Pnt } def +/Crs { stroke [] 0 setdash exch hpt sub exch vpt add M + hpt2 vpt2 neg V currentpoint stroke M + hpt2 neg 0 R hpt2 vpt2 V stroke } def +/TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M + hpt neg vpt -1.62 mul V + hpt 2 mul 0 V + hpt neg vpt 1.62 mul V closepath stroke + Pnt } def +/Star { 2 copy Pls Crs } def +/BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M + 0 vpt2 neg V hpt2 0 V 0 vpt2 V + hpt2 neg 0 V closepath fill } def +/TriUF { stroke [] 0 setdash vpt 1.12 mul add M + hpt neg vpt -1.62 mul V + hpt 2 mul 0 V + hpt neg vpt 1.62 mul V closepath fill } def +/TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M + hpt neg vpt 1.62 mul V + hpt 2 mul 0 V + hpt neg vpt -1.62 mul V closepath stroke + Pnt } def +/TriDF { stroke [] 0 setdash vpt 1.12 mul sub M + hpt neg vpt 1.62 mul V + hpt 2 mul 0 V + hpt neg vpt -1.62 mul V closepath fill} def +/DiaF { stroke [] 0 setdash vpt add M + hpt neg vpt neg V hpt vpt neg V + hpt vpt V hpt neg vpt V closepath fill } def +/Pent { stroke [] 0 setdash 2 copy gsave + translate 0 hpt M 4 {72 rotate 0 hpt L} repeat + closepath stroke grestore Pnt } def +/PentF { stroke [] 0 setdash gsave + translate 0 hpt M 4 {72 rotate 0 hpt L} repeat + closepath fill grestore } def +/Circle { stroke [] 0 setdash 2 copy + hpt 0 360 arc stroke Pnt } def +/CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def +/C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def +/C1 { BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 90 arc closepath fill + vpt 0 360 arc closepath } bind def +/C2 { BL [] 0 setdash 2 copy moveto + 2 copy vpt 90 180 arc closepath fill + vpt 0 360 arc closepath } bind def +/C3 { BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 180 arc closepath fill + vpt 0 360 arc closepath } bind def +/C4 { BL [] 0 setdash 2 copy moveto + 2 copy vpt 180 270 arc closepath fill + vpt 0 360 arc closepath } bind def +/C5 { BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 90 arc + 2 copy moveto + 2 copy vpt 180 270 arc closepath fill + vpt 0 360 arc } bind def +/C6 { BL [] 0 setdash 2 copy moveto + 2 copy vpt 90 270 arc closepath fill + vpt 0 360 arc closepath } bind def +/C7 { BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 270 arc closepath fill + vpt 0 360 arc closepath } bind def +/C8 { BL [] 0 setdash 2 copy moveto + 2 copy vpt 270 360 arc closepath fill + vpt 0 360 arc closepath } bind def +/C9 { BL [] 0 setdash 2 copy moveto + 2 copy vpt 270 450 arc closepath fill + vpt 0 360 arc closepath } bind def +/C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill + 2 copy moveto + 2 copy vpt 90 180 arc closepath fill + vpt 0 360 arc closepath } bind def +/C11 { BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 180 arc closepath fill + 2 copy moveto + 2 copy vpt 270 360 arc closepath fill + vpt 0 360 arc closepath } bind def +/C12 { BL [] 0 setdash 2 copy moveto + 2 copy vpt 180 360 arc closepath fill + vpt 0 360 arc closepath } bind def +/C13 { BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 90 arc closepath fill + 2 copy moveto + 2 copy vpt 180 360 arc closepath fill + vpt 0 360 arc closepath } bind def +/C14 { BL [] 0 setdash 2 copy moveto + 2 copy vpt 90 360 arc closepath fill + vpt 0 360 arc } bind def +/C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill + vpt 0 360 arc closepath } bind def +/Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto + neg 0 rlineto closepath } bind def +/Square { dup Rec } bind def +/Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def +/S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def +/S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def +/S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def +/S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def +/S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def +/S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill + exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def +/S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def +/S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill + 2 copy vpt Square fill + Bsquare } bind def +/S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def +/S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def +/S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill + Bsquare } bind def +/S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill + Bsquare } bind def +/S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def +/S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill + 2 copy vpt Square fill Bsquare } bind def +/S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill + 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def +/S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def +/D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def +/D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def +/D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def +/D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def +/D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def +/D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def +/D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def +/D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def +/D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def +/D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def +/D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def +/D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def +/D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def +/D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def +/D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def +/D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def +/DiaE { stroke [] 0 setdash vpt add M + hpt neg vpt neg V hpt vpt neg V + hpt vpt V hpt neg vpt V closepath stroke } def +/BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M + 0 vpt2 neg V hpt2 0 V 0 vpt2 V + hpt2 neg 0 V closepath stroke } def +/TriUE { stroke [] 0 setdash vpt 1.12 mul add M + hpt neg vpt -1.62 mul V + hpt 2 mul 0 V + hpt neg vpt 1.62 mul V closepath stroke } def +/TriDE { stroke [] 0 setdash vpt 1.12 mul sub M + hpt neg vpt 1.62 mul V + hpt 2 mul 0 V + hpt neg vpt -1.62 mul V closepath stroke } def +/PentE { stroke [] 0 setdash gsave + translate 0 hpt M 4 {72 rotate 0 hpt L} repeat + closepath stroke grestore } def +/CircE { stroke [] 0 setdash + hpt 0 360 arc stroke } def +/Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def +/DiaW { stroke [] 0 setdash vpt add M + hpt neg vpt neg V hpt vpt neg V + hpt vpt V hpt neg vpt V Opaque stroke } def +/BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M + 0 vpt2 neg V hpt2 0 V 0 vpt2 V + hpt2 neg 0 V Opaque stroke } def +/TriUW { stroke [] 0 setdash vpt 1.12 mul add M + hpt neg vpt -1.62 mul V + hpt 2 mul 0 V + hpt neg vpt 1.62 mul V Opaque stroke } def +/TriDW { stroke [] 0 setdash vpt 1.12 mul sub M + hpt neg vpt 1.62 mul V + hpt 2 mul 0 V + hpt neg vpt -1.62 mul V Opaque stroke } def +/PentW { stroke [] 0 setdash gsave + translate 0 hpt M 4 {72 rotate 0 hpt L} repeat + Opaque stroke grestore } def +/CircW { stroke [] 0 setdash + hpt 0 360 arc Opaque stroke } def +/BoxFill { gsave Rec 1 setgray fill grestore } def +/Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont +dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall +currentdict end definefont pop +/MFshow {{dup dup 0 get findfont exch 1 get scalefont setfont + [ currentpoint ] exch dup 2 get 0 exch rmoveto dup dup 5 get exch 4 get + {show} {stringwidth pop 0 rmoveto}ifelse 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a/macros/latex/contrib/pbsheet/xpl/xpl-fr.tex b/macros/latex/contrib/pbsheet/xpl/xpl-fr.tex new file mode 100644 index 0000000000..2b5e04e387 --- /dev/null +++ b/macros/latex/contrib/pbsheet/xpl/xpl-fr.tex @@ -0,0 +1,146 @@ +\documentclass{pbsheet} + +\TITRE{Feuille de TP n°1 -- Initiation à Matlab} +\FORMATION{Préparation à l'épreuve de modélisation + - Agrégation Externe de Mathématiques} +\ETABLISSEMENT{Université de l'Anonyme XXII} +\ANNEE{2004} +\MEL{agreg@labo.univ.fr} +\AUTEUR{A. Grégatif} +\WEB{http://www.labo.univ.fr/} +\DATE{\today} + +\begin{document} + +Ce TP porte sur les entrées et sorties, les fonctions et les outils +graphiques dont vous disposez sous Matlab. + +\section{Entrées et sorties} + +La commande \texttt{input} permet de demander à l'utilisateur Matlab d'entrer +les valeurs de variables à utiliser. La commande \texttt{pause} permet de +stopper l'exécution Matlab. Vous pouvez préciser le nombre de secondes de pose +ou revenir à Matlab en appuyant sur n'importe quelle touche. La commande +\texttt{save} permet de sauvegarder dans un fichier, dont le nom par défaut +est \texttt{matlab.mat}, le contenu de certaines variables dont vous souhaitez +garder une trace. Ce fichier peut être appelé par la commande \texttt{load} +qui restaure toutes les variables que vous avez sauvegardées. + +\begin{lstlisting}[language=Matlab]{Créations et Sauvegardes sous Matlab} +n=input('Entrez la valeur de n : '); % Affectez une valeur à n. +a=input('Precisez la valeur de a : '); % Affectez une valeur à a. +v=a.^[0:n]; A=toeplitz(v); d=det(A); % Création de la matrice de Toeplitz A +save restoep n a A d; % Sauvegarde de n, a, A, d dans restoep.mat +clear % Efface toutes les variables de la session. +load restoep % Restaure les variables de restoep.mat. +who % Vérification. +\end{lstlisting} + +\section{Fonctions} + +Un ensemble de commandes Matlab peut être considéré comme une fonction. On +peut voir une fonction comme un sous-programme Matlab dont les paramètres +éventuels sont les arguments de la fonction et dont les résultats sont les +images de cette fonction. Beaucoup de fonctions Matlab, comme par exemple +\texttt{mean}, sont déjà écrites en Matlab et le code Matlab correspondant est +stocké dans un fichier dont le nom se termine par \texttt{.m}. Pour +\texttt{mean}, il s'agit de \texttt{mean.m}. Ajouter de nouvelles fonctions à +Matlab revient donc à écrire de nouveaux fichiers de ce type. Il est d'usage +d'appeler une fonction du même nom que le fichier correspondant. + +\textbf{Simulation de lois discrètes.} Dans votre répertoire personnel, éditer +le fichier \texttt{probadis.m} suivant dont le code Matlab génère une +réalisation aléatoire d'une loi discrète à support fini. + +\MFILE{probadis} + +La commande Matlab \texttt{type} permet de lister le contenu d'un fichier. +Ainsi, \texttt{type probadis} vous montrera le code source Matlab de la +fonction \texttt{probadis}. Le commentaire ajouté à partir de la seconde ligne +constituera l'aide affiché lorsque l'utilisateur tapera \texttt{help + probadis}. Finalement, la commande \texttt{what} liste les fichiers Matlab +du répertoire courant. + +Voici une autre fonction de simulation de loi discrète, qui peut renvoyer une +matrice de réalisations. + +\MFILE{rdiscr} + +\begin{exo}[Loi binomiale] + Créer un code Matlab permettant de générer un vecteur aléatoire $X$ + contenant $N$ réalisations i.i.d. de loi binomiale $\cB(n,p)$ où les valeurs + $N,n\geq 1$ et $0<p<1$ sont affectées par l'utilisateur. Pour $N$ assez + grand, vérifier la LGN sur les moyennes empiriques successives de $X$. Voici + un exemple de programme qui fait l'affaire, et donc la sortie graphique se + trouve en page \pageref{fi:simbin}. +\end{exo} + +\MFILE{simbin} + +\begin{exo} + Soit $(X_n)_{n\in\dN}$ une suite de variables aléatoires i.i.d. de loi + exponentielle $\cE(\lambda)$ avec $\lambda >0$. Si $S_n=\sum_{k=1}^n X_k$, + $N_0=0$ et pour tout $t> 0$, $N_t=\sum_{n=1}^\infty \rI_\BRA{S_n\leq t}$, + $(N_t)_{t\geq0}$ est un processus de Poisson d'intensité $\lambda$. Montrer + que, pour tout $t>0$, $N_t$ suit la loi de Poisson $\cP(\la t)$. En déduire + un code Matlab permettant de générer un vecteur aléatoire $Y$ contenant $N$ + réalisations i.i.d. de loi $\cP(\la)$ où les valeurs $N\geq 1$ et $\la>0$ + sont affectées par l'utilisateur. Pour $N$ assez grand, vérifier la LGN sur + les moyennes empiriques successives de $Y$. +\end{exo} + +\begin{exo} + Pour $N$, $N_1$ et $n \geq 1$ avec $N_1,n \leq N$, la loi hypergéométrique + $\cH(N,N_1,n)$ est donnée, pour tout $k \in \dN$ avec $0\leq k \leq n$, par + $\dP(X=k)=C_{N_1}^kC_{N-N_1}^{n-k}/C_N^n$. Créer un code Matlab permettant + de générer un vecteur aléatoire $Z$ contenant $M$ réalisations i.i.d. de loi + $\cH(N,N_1,n)$ où les valeurs $M, N \geq 1$ et $N_1,n\leq N$ sont affectées + par l'utilisateur. + \begin{enumerate} + \item Si $N$ tend vers l'infini et le rapport $N_1/N$ tend vers $p$ avec + $0<p<1$, montrer que $X$ converge en loi vers la loi Binomiale $\cB(n,p)$. + Pour $M, N$ assez grand et $N_1=pN$ avec $0<p<1$, tracer l'histogramme de + $Z$ et comparer le à la loi $\cB(n,p)$. + \item Si $N$, $N_1$ et $n$ tendent vers l'infini et le produit $nN_1/N$ tend + vers $\la>0$, montrer que $X$ converge en loi vers la loi de Poisson + $\cP(\la)$. Pour $M$, $N$ assez grand, $N_1=\la\sqrt{N}$ et $n=\sqrt{N}$, + tracer l'histogramme de $Z$ et comparer le à la loi $\cP(\la)$. + \end{enumerate} +\end{exo} + +\section{Représentations graphiques} + +\begin{lstlisting}[language=Matlab] + {Illustration de la LGN pour la loi exponentielle} + clear; n=1000; lambda=0.5; X=-log(rand(n,1))/lambda; + figure; % Création d'une nouvelle fenêtre graphique. + plot(cumsum(X)'./[1:length(X)],'b') % Trace les moy. emp. successives de X. + title('Loi des Grands Nombres') % Titre de la figure. + xlabel('Nombre de réalisations') % Titre des abscisses. + ylabel('Moyennes empiriques') % Titre des ordonnées. + hold on % Garde la fenêtre graphique. + plot(1/lambda*ones(n,1),'r--'); % Trace la limite théorique. + legend('Empirique','Theorique'); % Légende. +\end{lstlisting} + +\begin{exo} + Ajouter à vos codes Matlab les représentations graphiques rencontrées + ci-dessus. +\end{exo} + +\FIG{simbin}{1.0}{htbp}{Sortie graphique du programme \ref{code:simbin} + \pageref{code:simbin}.} + +Voici une fonction Matlab qui permet de simuler la loi uniforme discrète finie +de façon rapide et simple. + +\MFILE{rint} + +\nocite{*} + +\phantomsection % Nécessaire pour le PDF. +\addcontentsline{toc}{section}{Bibliographie} +\bibliographystyle{amsalpha} +\bibliography{xpl-fr} + +\end{document} |