(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 1023805, 27350]*) (*NotebookOutlinePosition[ 1024735, 27380]*) (* CellTagsIndexPosition[ 1024691, 27376]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ (* This notebook contains procedures that simulate various \ processes in the Gott-Caves dispute. The basic organization of this notebook \ is as follows. In various comment cells, such as this one, we describe the \ simulations that the reader can perform. Below each such cell, there is one \ or more procedure that is automatically initialized the first time that any \ command in this notebook is entered. Examples of how these procedures are \ used follow the body of the procedure (in another cell). *)\ \>", "Text"], Cell[TextData[{ "(* This simulation consists of a sequence of busses, whose inter-arrival \ times are distributed according to one of the canned densities in ", StyleBox["Mathematica", FontSlant->"Italic"], ". By this we mean that we imagine that a sequence of values from this \ distribution is created, and these values define the times between arrivals \ of busses at a certain bus stop. This distribution is given the name cdist \ below, and can be chosen from the list given in th e statistical functions \ section of the ", StyleBox["Mathematica", FontSlant->"Italic"], " manual. We imagine that the process has been running for a long time, so \ the process is near `equilibrium.' A would-be passenger appears at the bus \ stop, at some specific time. The quantities that we will simulate are the \ time since the last bus arrived at the bus stop, the time until the next bus \ arrives at the bus stop, and the sum of these two times. We will call these \ quantities the age, the residual lifetime, and the lifetime, respectively, of \ the passenger's bus. We denote the density of the inter-arrival times of the \ busses by f(t) and the cumulative distribution function of the inter-arrival \ time by F(t). In renewal theory, it is shown that the density of both the \ age and the residual lifetime of the passenger's bus is equal to (1 - \ F(t))/m, where m is the mean of the inter-arrival distribution. Also, it is \ shown that the density of the passenger's bus is given by tf(t)/m. In the \ Gott-Caves dispute, it is important to understand the conditional \ distribution of the residual lifetime of the passenger's bus, given that the \ age of the bus equals some constant a. By this we mean that we want to \ simulate the probability that the residual lifetime is at most y, given that \ the age equals a. In the Chance News article, we explain why this is equal to\ \n\nF(y + a)/(1 - F(a)). \n\nFinally, Gott's main thesis is that if A \ represents the age of the passenger's bus, R represents the residual age of \ the passenger's bus, and c is a positive constant, then the probability that \ R exceeds cA equals 1/(1+c). *)" }], "Text"], Cell[BoxData[ \(<< Statistics`ContinuousDistributions`\)], "Input", InitializationCell->True], Cell["\<\ (* The above command is automatically executed when the user \ answers yes to the question asking whether the user wants all of the \ initialization cells entered. This question is asked the first time any \ command is entered, once this notebook is open. For example, to get this \ question to appear on the screen, one can put the cursor on the above \ command, and hit the `enter' key. *)\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(cdist\ = \ ExponentialDistribution[1]\)], "Input"], Cell[BoxData[ \(ExponentialDistribution[1]\)], "Output"] }, Open ]], Cell["\<\ PROGRAM: Areabargraph CALLING SEQUENCE: Areabargraph[data, xmin, xmax, k] PARAMETERS: data - a list of real numbers xmin, xmax - real numbers k - an integer SYNOPSIS: - This program divides the interval [xmin, xmax] on the x-axis into k \ subintervals of equal length. Then, on each subinterval, a box is erected whose \ area is equal to the percentage of values in the list data which are in the \ subinterval. Important note: only data values which fall in the user-defined \ interval will be included in\tthe bar graph. If not all data values are included, a \ warning is displayed. I f you wish to be sure of including all data values in \ the graph, choose xmin = Min[data] and xmax = Max[data]. (See also \"Bargraph.\ \") RETURNED VALUES: - none LOCATION: File: \"Important Programs\"\ \>", "Text", CellOpen->False], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[Areabargraph];\)\), "\[IndentingNewLine]", \(Areabargraph[data_, xmin_, xmax_, k_] := Block[{sorteddata = {}, length = Length[data], dx = \((xmax - xmin)\)/k, currentupperlim = xmin + dx, result = {}, index = 1, counter, rectanglelist = {}, linelist = {}, graphicslist = {}}, sorteddata = Sort[data]; \[IndentingNewLine]If[\((\((sorteddata[\([1]\)] < xmin)\) || \((sorteddata[\([length]\)] > xmax)\))\), Print["\"]]; \[IndentingNewLine]Block[{}, For[i = 1, i \[LessEqual] k, \(i++\), counter = 0; \[IndentingNewLine]While[\((\((index \[LessEqual] length)\) && \((sorteddata[\([index]\)] \[LessEqual] currentupperlim)\))\), \(counter++\); \ \[IndentingNewLine]\(index++\);]; \[IndentingNewLine]result = Append[result, counter]; \[IndentingNewLine]currentupperlim += dx;]; \[IndentingNewLine]rectanglelist = Table[Rectangle[{xmin + \((i - 1)\)*dx, 0}, {xmin + i*dx, result[\([i]\)]/\((length*dx)\)}], {i, 1, k}]; \[IndentingNewLine]rectanglelist = Prepend[rectanglelist, RGBColor[0, 0, 1]]; \[IndentingNewLine]linelist = Table[{Line[{{xmin + \((i - 1)\)*dx, 0}, {xmin + \((i - 1)\)*dx, result[\([i]\)]/\((length*dx)\)}}], Line[{{xmin + i*dx, 0}, {xmin + i*dx, result[\([i]\)]/\((length*dx)\)}}], Line[{{xmin + \((i - 1)\)*dx, result[\([i]\)]/\((length*dx)\)}, {xmin + i*dx, result[\([i]\)]/\((length*dx)\)}}]}, {i, 1, k}]; \[IndentingNewLine]linelist = Flatten[linelist, 1]; \[IndentingNewLine]graphicslist = Union[rectanglelist, linelist]; \[IndentingNewLine]Show[ Graphics[{rectanglelist, linelist}, Axes \[Rule] True, PlotRange \[Rule] All, Frame \[Rule] True]]]]\)}], "Input", CellOpen->False, InitializationCell->True], Cell[BoxData[ \(General::"spell1" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(length\)\" is similar \ to existing symbol \"\!\(Length\)\"."\)], "Message"] }, Open ]], Cell[TextData[{ "(* The following set of commands give basic examples of how one can use ", StyleBox["Mathematica", FontSlant->"Italic"], " to generate a set of random values from the distribution, put them in a \ histogram, and plot the theoretical density with the histogram. *)" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[Random[cdist], \ {100}]\)], "Input"], Cell[BoxData[ \({0.11659850096815523`, 0.6078279632278878`, 0.17773383307385202`, 0.048558612655784573`, 0.046868512823143645`, 0.9232416660084927`, 0.6727788150129582`, 0.5382088994601022`, 0.8416627619477111`, 0.2595674012912986`, 1.0768508963860448`, 0.0961813660409301`, 1.4890897384386297`, 1.8276679354822245`, 3.918541765042834`, 0.5925222134826804`, 1.5726747625582822`, 0.09458550472522048`, 0.010608527379679851`, 0.03793983105403181`, 0.832854800911434`, 0.06826209287021398`, 1.6153269266784693`, 0.3138875122114063`, 0.6072194327271695`, 0.9429342569820321`, 1.0170495215325002`, 0.2510291351561058`, 0.5265300637410327`, 0.007776109932515885`, 0.1609063119546392`, 1.6388319363768762`, 1.834723437485512`, 1.5101862744246226`, 0.6719627672659736`, 1.2520869847652467`, 0.06819345469649087`, 2.8120660744958013`, 0.7116469527054222`, 0.31064153596675564`, 0.31939250976902256`, 1.8949171501044746`, 0.6903758152557914`, 0.2610976485598671`, 1.2317412299577546`, 1.5310189218133798`, 1.1954692418435942`, 3.22881255067406`, 0.29179682405739443`, 0.19015394940129698`, 0.06091681302659472`, 1.340920329568845`, 1.8561742364525444`, 0.18082932502501642`, 2.4131851272580334`, 2.6971393501580665`, 0.003394066571827136`, 0.4882344759648893`, 0.54675599318288`, 0.24655222105958569`, 2.7720779282489403`, 0.5912618944586199`, 2.430539728244891`, 3.025903364092012`, 1.0908178446989165`, 0.9080809397598891`, 0.5334091819482503`, 1.2790258975598516`, 3.1199911024466855`, 1.6767413875674266`, 1.258640984597564`, 1.4325319518500497`, 1.2132245213720645`, 1.02123340899768`, 1.0696217891893338`, 0.02316732717857692`, 1.9592227422648363`, 0.6432668199777364`, 1.3719631652421478`, 0.0946393758423562`, 1.9354710068827854`, 0.09226474190925357`, 0.39336246685730647`, 2.0540827295950184`, 2.503180680600204`, 1.026564727600537`, 0.5330801166354395`, 2.5295307435891883`, 0.2931869377976106`, 0.04610880790414606`, 8.552498952499738`, 0.2214098795625678`, 0.3542135005156435`, 0.2640231457381047`, 0.33386105212617445`, 0.5750364291208168`, 0.9051335845657869`, 0.8969653967842649`, 0.9861366140179585`, 0.5351432560971591`}\)], "Output"] }, Open ]], Cell["\<\ (* It is sometimes helpful, for what follows, to know the maximum \ value in such a list. 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00000040oooo000>0?ooo`030000003oool0oooo00@0oooo00<000000?ooo`3oool00`3oool00`00 0000oooo0?ooo`0=0?ooo`030000003oool0oooo00d0oooo00<000000?ooo`3oool03@3oool00`00 0000oooo0?ooo`0<0?ooo`030000003oool0oooo00d0oooo00<000000?ooo`3oool03@3oool00`00 0000oooo0?ooo`0=0?ooo`030000003oool0oooo00d0oooo00<000000?ooo`3oool0303oool00`00 0000oooo0?ooo`0=0?ooo`030000003oool0oooo00d0oooo00<000000?ooo`3oool03@3oool00`00 0000oooo0?ooo`0=0?ooo`030000003oool0oooo00`0oooo00<000000?ooo`3oool03@3oool00`00 0000oooo0?ooo`0=0?ooo`030000003oool0oooo00<0oooo0@0000010?ooo`005@3ooooo000000/0 00000@3oool00001\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.877277, -0.103856, \ 0.0316508, 0.00640152}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell["\<\ (* Now we wish to simulate the various quantities associated with \ the bus problem. To make the process close to equilibrium, we allow the user \ to put in a time, arrivet, that the passenger arrives at the bus stop. \ Presumably, the larger this time is (in relation to the mean of the \ interarrival times of the busses, the closer the process will be to \ equilibrium. It should be pointed out that it is also true that the larger \ this time is, the longer this program takes to generate the required number \ of data values. The number n, given by the user, is the desired size of the \ data set. The output lists are called lastbuslist, nextbuslist, and \ passengerbustimeslist. *)\ \>", "Text"], Cell[BoxData[ \(Bussimulation[cdist_, \ n_, \ arrivet_]\ := \[IndentingNewLine]Block[{lastbuslist\ = \ {}, \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ nextbuslist\ = \ {}, \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ passengerbustimeslist\ = \ \ {}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ timesubtotal, \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ currbustime\[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ }, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ For[ i\ = \ 1, \ i\ \[LessEqual] \ n, \ \(i++\), \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ timesubtotal\ = \ 0; \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ While[timesubtotal\ < \ arrivet, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ currbustime\ = \ Random[cdist]; \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ timesubtotal\ = \ timesubtotal\ + \ currbustime;\[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ]; \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ lastbuslist\ = \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Append[ lastbuslist, \ \ \ \ arrivet\ - \ \((timesubtotal\ - \ currbustime)\)]; \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ nextbuslist\ = \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Append[nextbuslist, \ timesubtotal\ - \ arrivet]; \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ passengerbustimeslist\ = \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Append[passengerbustimeslist, \ currbustime];\[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ]; \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ Return[{lastbuslist, \ nextbuslist, \ passengerbustimeslist}]\[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ ]\)], "Input", InitializationCell->True], Cell["\<\ (* The following procedure displays the output of the above \ procedure Bussimulation, along with the theoretical densities given by \ renewal theory. The first two lists output by the above program should be \ distributed according to the inter-arrival density f(t). The user should \ define this density BEFORE using this procedure. The variable inputlists is \ just the set of three lists output by the above procedure Bussimulation. The \ variables c1 and c2 are the endpoints of the interval on which the density is \ defined (c2 can equal Infinity) and c3 and c4 are the endpoints over which \ the lists will be graphed (both c3 and c4 must be finite). The third list \ output by the above program is distributed according to the density t f(t)/m, \ where m is equal to the mean of the density (this is calculated by the \ procedure). \ \>", "Text"], Cell[BoxData[ \(bussimulationplot[inputlists_, f_, \ c1_, \ c2_, \ c3_, \ c4_]\ := \[IndentingNewLine]Block[{lastbuslist\ = \ inputlists[\([1]\)], \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ nextbuslist\ = \ inputlists[\([2]\)], \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ passengerbuslist\ = \ inputlists[\([3]\)], \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ m, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ }, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ m\ = \ Integrate[t\ f[t], \ {t, \ c1, \ c2}]; \[IndentingNewLine]Show[ Areabargraph[lastbuslist, c3, \ c4, 20], \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ Plot[\((1\ - Integrate[f[s], \ {s, c1, t}])\)/m, \ {t, \ c3, c4}]\[IndentingNewLine]\ \ \ \ \ \ \ \ ]; \ \[IndentingNewLine]Show[ Areabargraph[nextbuslist, c3, \ c4, \ 20], \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ Plot[\((1\ - Integrate[f[s], \ {s, c1, t}])\)/m, \ {t, \ c3, c4}]\[IndentingNewLine]\ \ \ \ \ \ \ \ ]; \ \[IndentingNewLine]Show[Areabargraph[passengerbuslist, c3, \ c4, \ 20], \ Plot[t*f[t]/m, \ {t, \ c3, \ c4}]];\[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ ]\)], \ "Input"], Cell["\<\ (* The first distribution we will try is the exponential density \ with parameter lambda = 1. *)\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(cdist\ = \ ExponentialDistribution[1]\)], "Input"], Cell[BoxData[ \(ExponentialDistribution[1]\)], "Output"] }, Open ]], Cell[BoxData[ \(f[t_]\ := \ E^\((\(-t\))\)\)], "Input"], Cell["\<\ (* The following set of calculations are for an arrival time of 50. \ *)\ \>", "Text"], Cell[BoxData[ \(\(Bussimulation[cdist, \ 10000, \ 50];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(bussimulationplot[%40, \ f, \ 0, \ Infinity, \ 0, \ 10]\)], "Input"], Cell[BoxData[ \("Note: some data values lie outside the user-defined 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