(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 56360, 1003] NotebookOptionsPosition[ 55560, 974] NotebookOutlinePosition[ 55902, 989] CellTagsIndexPosition[ 55859, 986] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Project - Random Walk", "Title", CellChangeTimes->{{3.451688698491304*^9, 3.45168870388547*^9}}], Cell["\<\ The graph below show a walk of 100 steps. Each step is 1 unit long but its \ direction is choosen at random. This particular random walk starts at the \ origin (the black dot) and ends at the red dot with coordinates \ (2.42851,12.6146)\ \>", "Text", CellChangeTimes->{{3.451688742308855*^9, 3.451688867028472*^9}, 3.4516889645026073`*^9, {3.451689023579369*^9, 3.451689104200831*^9}, { 3.4516892131295547`*^9, 3.451689216392599*^9}, {3.451689664846039*^9, 3.451689665785678*^9}}], Cell[BoxData[ GraphicsBox[{ LineBox[{{0, 0}, {0.8658894565054281, -0.500235393702539}, { 1.5935433223585709`, 0.18570910223643744`}, {0.9238190905999815, 0.9283189953804536}, {1.8792690990708745`, 1.2234719748893355`}, { 2.792502598680975, 1.6309085652808815`}, {3.6892318373846487`, 2.073488131521721}, {3.342257855777115, 3.0113628865329725`}, { 3.7031703539867724`, 2.0787632035842467`}, {3.1219723908692467`, 2.892525409433129}, {2.2000147843225024`, 2.5052345991489204`}, { 3.0243011233446357`, 1.9390614531048072`}, {2.0495058749392205`, 1.715960077747718}, {1.8123998725475199`, 2.687443862290634}, { 2.69336445124614, 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It ends at (-6.54248, \ -10.4413).\ \>", "Text", CellChangeTimes->{{3.451689168186178*^9, 3.451689187262488*^9}, { 3.451689234259552*^9, 3.451689252269616*^9}}], Cell[BoxData[ GraphicsBox[{ LineBox[{{0, 0}, { 0.43181961474103836`, -0.901959988206185}, {-0.06597165437542885, \ -1.7692568534094537`}, {-0.5574663531424001, -0.8983762869228865}, \ {-1.5568773536875402`, -0.8640593524828494}, {-1.865750345240644, 0.08704394111308611}, {-1.8553271445135466`, -0.9129017358547193}, \ {-2.8079273734892545`, -1.2171266588236125`}, {-3.7957460704206145`, \ -1.061517462460319}, {-3.59751915576572, -2.0416736185428084`}, \ {-4.120921301253571, -2.89375940888244}, {-4.512339308543657, \ -1.9735464207274926`}, {-4.504824263068429, -2.973518182374543}, \ {-3.578647456238388, -2.596428640633979}, {-4.249704924959401, \ -1.855023303673321}, {-4.302981713268871, -0.8564435202671938}, \ {-4.170212434187178, -1.84759049147385}, {-4.239844664877426, \ -0.8500177610681415}, {-3.984967196017285, 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only, \ you get something like this:\ \>", "Text", CellChangeTimes->{{3.4516892665404873`*^9, 3.451689312934533*^9}, { 3.4516894798726883`*^9, 3.4516894827740183`*^9}, 3.4516899794546337`*^9}], Cell[BoxData[ GraphicsBox[ {RGBColor[1, 0, 0], PointSize[0.01], PointBox[CompressedData[" 1:eJwVl3c8lu8Xx+29nsfez+OxyUxF5T4iZSSESkPISKVtJBnJKFTyLZoSSkui ELlPKlmprERZ2Xvv8fP76/rnvl73db3Oud7n/bE/4Ub3YGJiYmFiYu39/0ou tvAu2SwcAvFrOb/f5W0D5TXNjJMq74Dzd8Mvvz0ZxLIUyxMr998g9/uJgMtu B5DzN+a6khCHlEpXtV0lUUifbTdyW/pIDuK1jH1CO1DhHtPmuXeCIOoc/oTn yRgqVd0ysC1dBnpKVafuJkAJvRkzc9lgVP25c68Nmz1wh5J5z9UjkGlXy560 vb0guNff/qphE9nrtubu22IPoH3hf9O/bgZ4drkeLt5gDHTBJvLa6adAS94w LNJFkO1mhRo2wduA9d5N3a3z4ki3OzvjbfYJOH/NH7ba8BAUX0UH2VTxIOWb g3+pfBEoJNWt29rYR3z99oXzXeY9oJ35EPEj15qcuObdG1XRj1xN/sN21anA vE719sUrn5HdQqrL6qoYSvOf310iqAhqF3j6G07sQI6eY4H2fvGo9idQpCng ATEzLadKshiCTv7lxi891iDq9/nkW4UfqHoooX/qdSm5kjJ+0M9TB9U2/RUY P98D9KOcYeIKDJRTz7URKvmCUoXG4lXeTUgLv/WAa9tFoFbkreXO20G8nRuf 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The distribution of these distances among the 1000 random walks is best \ shown with a histogram:" }], "Text", CellChangeTimes->{{3.451689498938498*^9, 3.451689616070779*^9}, { 3.451689705215578*^9, 3.4516897895366907`*^9}, {3.451689826116523*^9, 3.4516898898537617`*^9}}], Cell[BoxData[ GraphicsBox[{ {RGBColor[0.7771114671549554, 0.7981689173723965, 0.92304875257496], EdgeForm[GrayLevel[0]], RectangleBox[{0.5, 0}, {1., 8}]}, {RGBColor[0.7771114671549554, 0.7981689173723965, 0.92304875257496], EdgeForm[GrayLevel[0]], RectangleBox[{1., 0}, {1.5, 13}]}, {RGBColor[0.7771114671549554, 0.7981689173723965, 0.92304875257496], EdgeForm[GrayLevel[0]], RectangleBox[{1.5, 0}, {2., 15}]}, {RGBColor[0.7771114671549554, 0.7981689173723965, 0.92304875257496], EdgeForm[GrayLevel[0]], RectangleBox[{2., 0}, {2.5, 14}]}, {RGBColor[0.7771114671549554, 0.7981689173723965, 0.92304875257496], EdgeForm[GrayLevel[0]], RectangleBox[{2.5, 0}, {3., 28}]}, {RGBColor[0.7771114671549554, 0.7981689173723965, 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PlotLabel->None, PlotRange->{{0.5, 24.}, All}, PlotRangeClipping->True, PlotRangePadding->{ Scaled[0.02], Scaled[0.02]}, PlotRegion->Automatic, PreserveImageOptions->Automatic, Prolog->{}, RotateLabel->True, Ticks->{{{5., FormBox["5", TraditionalForm]}, {10., FormBox["10", TraditionalForm]}, {15., FormBox["15", TraditionalForm]}, {20., FormBox["20", TraditionalForm]}}, Automatic}, TicksStyle->{}]], "Output", CellChangeTimes->{{3.4516885840474367`*^9, 3.451688623714438*^9}, { 3.451689453331625*^9, 3.451689455981255*^9}, {3.451690139408642*^9, 3.451690163742324*^9}}], Cell["\<\ This shows that the most likely distance travelled in a random walk of 100 \ steps is about 9 Indeed the average distance over all 1000 random walks is \ 9.00895.\ \>", "Text", CellChangeTimes->{{3.451689909179755*^9, 3.451689939428968*^9}, { 3.45169021278553*^9, 3.451690259620723*^9}}], Cell[CellGroupData[{ Cell["Task", "Subsection", CellChangeTimes->{{3.420156478692501*^9, 3.42015647912057*^9}}], Cell[TextData[{ "Your job is to develop ", StyleBox["Mathematica", FontSlant->"Italic"], " programs to produce graphs like the above and to investigate how the \ average distance depends on the number of steps in the random walk. The main \ question to be resolved is, if the number of steps is doubled from 100 to \ 200, does the average distance traveled double too?\n\nTo make the histogram, \ you can load in the Histogram package and use the ", StyleBox["Histogram", "Input"], " command. See the documentation to find out how this works. Other useful \ commands that you might not know are ", StyleBox["Norm", "Input"], " and ", StyleBox["Mean", "Input"], "." }], "Text", CellChangeTimes->{{3.451690362044073*^9, 3.4516904395046186`*^9}, { 3.4516906714476557`*^9, 3.451690866489463*^9}, {3.451690903361326*^9, 3.45169096865679*^9}, {3.451691024839282*^9, 3.451691055085376*^9}, 3.451691349044018*^9}] }, Open ]] }, Open ]] }, WindowSize->{795, 832}, WindowMargins->{{283, Automatic}, {Automatic, 26}}, FrontEndVersion->"6.0 for Mac OS X PowerPC (32-bit) (April 20, 2007)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[590, 23, 103, 1, 76, "Title"], Cell[696, 26, 502, 9, 41, "Text"], Cell[1201, 37, 5033, 78, 370, "Output"], Cell[6237, 117, 221, 5, 26, "Text"], Cell[6461, 124, 4738, 75, 370, "Output"], Cell[11202, 201, 278, 5, 26, "Text"], Cell[11483, 208, 34261, 566, 374, 21160, 349, "CachedBoxData", "BoxData", \ "Output"], Cell[45747, 776, 669, 17, 67, "Text"], Cell[46419, 795, 7762, 140, 244, "Output"], Cell[54184, 937, 301, 6, 41, "Text"], Cell[CellGroupData[{ Cell[54510, 947, 91, 1, 34, "Subsection"], Cell[54604, 950, 928, 20, 101, "Text"] }, Open ]] }, Open ]] } ] *) (* End of internal cache information *)