(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 11.2' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 59565, 1409] NotebookOptionsPosition[ 50551, 1238] NotebookOutlinePosition[ 53024, 1296] CellTagsIndexPosition[ 52549, 1282] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{"Student", " ", "Name"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ RowBox[{"July", " ", "24"}], ",", " ", "2018"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ RowBox[{"Physics", " ", "III"}], " ", "-", " ", RowBox[{"Introduction", " ", "to", " ", "Statistical", " ", "Physics"}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"(*", RowBox[{" ", RowBox[{ RowBox[{"Assignment", " ", "5"}], " ", "-", " ", RowBox[{"Problem", " ", "5"}]}], ")"}]}]}]], "Input", CellChangeTimes->{{3.740827998671999*^9, 3.7408280624187*^9}, { 3.740828148309654*^9, 3.740828149213911*^9}, {3.740828182060598*^9, 3.740828198285409*^9}, {3.7408283554163*^9, 3.740828362438055*^9}, { 3.740828407788045*^9, 3.7408284323215*^9}, {3.7408301476194963`*^9, 3.7408301631292467`*^9}, {3.740831263315872*^9, 3.74083130330422*^9}, { 3.740832047143058*^9, 3.740832060574497*^9}, {3.7408321081590223`*^9, 3.7408321337623034`*^9}, {3.740832742145514*^9, 3.740832765441086*^9}, 3.7408333314702053`*^9, 3.7408337601410923`*^9, {3.740833858916465*^9, 3.7408338624027157`*^9}, {3.740837327076077*^9, 3.7408373324560204`*^9}, { 3.740838912335972*^9, 3.740838915761477*^9}, {3.741050704519249*^9, 3.741050708820043*^9}},ExpressionUUID->"0ddec9cd-366e-4c71-a200-\ dfc298dafbea"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["(a)", FontWeight->"Bold"], " Mean and Standard Deviation of Binomial Distribution" }], "Subsubsection", CellChangeTimes->{{3.740832790450245*^9, 3.740832798281765*^9}, { 3.740835278343739*^9, 3.740835281630425*^9}, {3.740835324474875*^9, 3.740835340112845*^9}, {3.740837302843678*^9, 3.74083730503259*^9}},ExpressionUUID->"27770dff-b8c6-4028-a08b-\ 43a423cf3356"], Cell[TextData[{ "For the binomial distribution where the probability of getting heads is \ 50%, the mean for N coin flips is ", StyleBox["< k> = N/2", FontWeight->"Bold"], " and the standard deviation is ", StyleBox["\[Sigma]_k= sqrt(N)/4", FontWeight->"Bold"], ". If we take N = 1000, we have ", StyleBox[" = 500", FontWeight->"Bold"], " and ", StyleBox["\[Sigma]_k = 15.811. 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\\\"TI\\\"]\\), then repeatedly \ evaluates \\!\\(\\*StyleBox[\\\"body\\\", \\\"TI\\\"]\\) and \ \\!\\(\\*StyleBox[\\\"incr\\\", \\\"TI\\\"]\\) until \ \\!\\(\\*StyleBox[\\\"test\\\", \\\"TI\\\"]\\) fails to give True. \"\>", "\[NonBreakingSpace]", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/For"]}]], "Print", "PrintUsage", CellChangeTimes->{3.741049224921628*^9}, CellTags-> "Info13741034823-4460184",ExpressionUUID->"e73956d0-cfcb-430f-9598-\ a8541838b4ea"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "RandomReal"}]], "Input", CellChangeTimes->{{3.7408329889031076`*^9, 3.7408329937035713`*^9}, { 3.740833640136971*^9, 3.7408336408149548`*^9}},ExpressionUUID->"8794e215-cef7-464b-95b9-\ b709c1ee2880"], Cell[BoxData[ RowBox[{"\<\"\\!\\(\\*RowBox[{\\\"RandomReal\\\", \\\"[\\\", \\\"]\\\"}]\\) \ gives a pseudorandom real number in the range 0 to 1. \ \\n\\!\\(\\*RowBox[{\\\"RandomReal\\\", \\\"[\\\", RowBox[{\\\"{\\\", \ RowBox[{SubscriptBox[StyleBox[\\\"x\\\", \\\"TI\\\"], StyleBox[\\\"min\\\", \ \\\"TI\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"x\\\", \\\"TI\\\"], \ StyleBox[\\\"max\\\", \\\"TI\\\"]]}], \\\"}\\\"}], \\\"]\\\"}]\\) gives a \ pseudorandom real number in the range \ \\!\\(\\*SubscriptBox[StyleBox[\\\"x\\\", \\\"TI\\\"], StyleBox[\\\"min\\\", \ \\\"TI\\\"]]\\) to \\!\\(\\*SubscriptBox[StyleBox[\\\"x\\\", \\\"TI\\\"], \ StyleBox[\\\"max\\\", \\\"TI\\\"]]\\). \ \\n\\!\\(\\*RowBox[{\\\"RandomReal\\\", \\\"[\\\", \ SubscriptBox[StyleBox[\\\"x\\\", \\\"TI\\\"], StyleBox[\\\"max\\\", \ \\\"TI\\\"]], \\\"]\\\"}]\\) gives a pseudorandom real number in the range 0 \ to \\!\\(\\*SubscriptBox[StyleBox[\\\"x\\\", \\\"TI\\\"], \ StyleBox[\\\"max\\\", \\\"TI\\\"]]\\).\\n\\!\\(\\*RowBox[{\\\"RandomReal\\\", \ \\\"[\\\", RowBox[{StyleBox[\\\"range\\\", \\\"TI\\\"], \\\",\\\", \ StyleBox[\\\"n\\\", \\\"TI\\\"]}], \\\"]\\\"}]\\) gives a list of \ \\!\\(\\*StyleBox[\\\"n\\\", \\\"TI\\\"]\\) pseudorandom reals. \ \\n\\!\\(\\*RowBox[{\\\"RandomReal\\\", \\\"[\\\", RowBox[{StyleBox[\\\"range\ \\\", \\\"TI\\\"], \\\",\\\", RowBox[{\\\"{\\\", \ RowBox[{SubscriptBox[StyleBox[\\\"n\\\", \\\"TI\\\"], StyleBox[\\\"1\\\", \ \\\"TR\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"n\\\", \\\"TI\\\"], \ StyleBox[\\\"2\\\", \\\"TR\\\"]], \\\",\\\", StyleBox[\\\"\[Ellipsis]\\\", \\\ \"TR\\\"]}], \\\"}\\\"}]}], \\\"]\\\"}]\\) gives an \ \\!\\(\\*SubscriptBox[StyleBox[\\\"n\\\", \\\"TI\\\"], StyleBox[\\\"1\\\", \\\ \"TR\\\"]]\\)\[Times]\\!\\(\\*SubscriptBox[StyleBox[\\\"n\\\", \\\"TI\\\"], \ StyleBox[\\\"2\\\", \\\"TR\\\"]]\\)\[Times]\[Ellipsis] array of pseudorandom \ reals. \"\>", "\[NonBreakingSpace]", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/RandomReal"]}]], "Print", "PrintUsage", CellChangeTimes->{3.741049228503714*^9}, CellTags-> "Info23741034828-4460184",ExpressionUUID->"89d6054d-5843-4e8c-a3e5-\ 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\\!\\(\\*StyleBox[\\\"condition\\\", \\\"TI\\\"]\\) evaluates to neither True \ nor False. \"\>", "\[NonBreakingSpace]", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/If"]}]], "Print", "PrintUsage", CellChangeTimes->{3.741049231159741*^9}, CellTags-> "Info33741034831-4460184",ExpressionUUID->"aa0c85e8-ce4e-457c-ba70-\ e1640a8f5a8b"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "Print"}]], "Input",ExpressionUUID->"f5b3bc30-635c-4f49-af65-39d16825acd4"], Cell[BoxData[ RowBox[{"\<\"\\!\\(\\*RowBox[{\\\"Print\\\", \\\"[\\\", \ StyleBox[\\\"expr\\\", \\\"TI\\\"], \\\"]\\\"}]\\) prints \ \\!\\(\\*StyleBox[\\\"expr\\\", \\\"TI\\\"]\\) as output. \"\>", "\[NonBreakingSpace]", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/Print"]}]], "Print", "PrintUsage", CellChangeTimes->{3.741049232913701*^9}, CellTags-> "Info43741034832-4460184",ExpressionUUID->"b171c249-8def-4b59-8779-\ 700b4c485392"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "Range"}]], "Input", CellChangeTimes->{{3.740833001436778*^9, 3.740833002193665*^9}, { 3.740833584376494*^9, 3.740833591424643*^9}},ExpressionUUID->"98640a8d-17af-466f-ba36-\ c4af7a8c5192"], Cell[BoxData[ RowBox[{"\<\"\\!\\(\\*RowBox[{\\\"Range\\\", \\\"[\\\", \ SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], StyleBox[\\\"max\\\", \ \\\"TI\\\"]], \\\"]\\\"}]\\) generates the list \\!\\(\\*RowBox[{\\\"{\\\", \ RowBox[{\\\"1\\\", \\\",\\\", \\\"2\\\", \\\",\\\", \ StyleBox[\\\"\[Ellipsis]\\\", \\\"TR\\\"], \\\",\\\", \ SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], StyleBox[\\\"max\\\", \ \\\"TI\\\"]]}], \\\"}\\\"}]\\). \\n\\!\\(\\*RowBox[{\\\"Range\\\", \\\"[\\\", \ RowBox[{SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], StyleBox[\\\"min\\\", \ \\\"TI\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], \ StyleBox[\\\"max\\\", 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\ a histogram with bin width specification \\!\\(\\*StyleBox[\\\"bspec\\\", \ \\\"TI\\\"]\\).\\n\\!\\(\\*RowBox[{\\\"Histogram\\\", \\\"[\\\", \ RowBox[{RowBox[{\\\"{\\\", RowBox[{SubscriptBox[StyleBox[\\\"x\\\", \ \\\"TI\\\"], StyleBox[\\\"1\\\", \\\"TR\\\"]], \\\",\\\", \ SubscriptBox[StyleBox[\\\"x\\\", \\\"TI\\\"], StyleBox[\\\"2\\\", \ \\\"TR\\\"]], \\\",\\\", StyleBox[\\\"\[Ellipsis]\\\", \\\"TR\\\"]}], \\\"}\\\ \"}], \\\",\\\", StyleBox[\\\"bspec\\\", \\\"TI\\\"], \\\",\\\", \ StyleBox[\\\"hspec\\\", \\\"TI\\\"]}], \\\"]\\\"}]\\) plots a histogram with \ bin heights computed according to the specification \ \\!\\(\\*StyleBox[\\\"hspec\\\", \ \\\"TI\\\"]\\).\\n\\!\\(\\*RowBox[{\\\"Histogram\\\", \\\"[\\\", \ RowBox[{RowBox[{\\\"{\\\", RowBox[{SubscriptBox[StyleBox[\\\"data\\\", \\\"TI\ \\\"], StyleBox[\\\"1\\\", \\\"TR\\\"]], \\\",\\\", \ SubscriptBox[StyleBox[\\\"data\\\", \\\"TI\\\"], StyleBox[\\\"2\\\", \\\"TR\\\ \"]], \\\",\\\", StyleBox[\\\"\[Ellipsis]\\\", \\\"TR\\\"]}], \\\"}\\\"}], \\\ \",\\\", StyleBox[\\\"\[Ellipsis]\\\", \\\"TR\\\"]}], \\\"]\\\"}]\\) plots \ histograms for multiple datasets \\!\\(\\*SubscriptBox[StyleBox[\\\"data\\\", \ \\\"TI\\\"], StyleBox[\\\"i\\\", \\\"TI\\\"]]\\).\"\>", "\[NonBreakingSpace]", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/Histogram"]}]], "Print", "PrintUsage", CellChangeTimes->{3.741049240504341*^9}, CellTags-> "Info73741034840-4460184",ExpressionUUID->"42f490a8-623e-44ae-a460-\ d49d7f73dd69"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "Mean"}]], "Input", CellChangeTimes->{{3.740833009716514*^9, 3.740833011790655*^9}, { 3.740833604202251*^9, 3.74083361296806*^9}},ExpressionUUID->"9029496f-42c9-475b-b060-\ 5d39cdc8f8cf"], Cell[BoxData[ RowBox[{"\<\"\\!\\(\\*RowBox[{\\\"Mean\\\", \\\"[\\\", \ StyleBox[\\\"list\\\", \\\"TI\\\"], \\\"]\\\"}]\\) gives the statistical mean \ of the elements in \\!\\(\\*StyleBox[\\\"list\\\", \\\"TI\\\"]\\). \ \\n\\!\\(\\*RowBox[{\\\"Mean\\\", \\\"[\\\", StyleBox[\\\"dist\\\", \ \\\"TI\\\"], \\\"]\\\"}]\\) gives the mean of the symbolic distribution \ \\!\\(\\*StyleBox[\\\"dist\\\", \\\"TI\\\"]\\).\"\>", "\[NonBreakingSpace]", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/Mean"]}]], "Print", "PrintUsage", CellChangeTimes->{3.7410492450234623`*^9}, CellTags-> "Info83741034844-4460184",ExpressionUUID->"bc951557-ff9a-46af-9c9c-\ 06df118ace1f"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "StandardDeviation"}]], "Input",ExpressionUUID->"47116c2e-7979-4280-b05f-c6668f680605"], Cell[BoxData[ RowBox[{"\<\"\\!\\(\\*RowBox[{\\\"StandardDeviation\\\", \\\"[\\\", \ StyleBox[\\\"list\\\", \\\"TI\\\"], \\\"]\\\"}]\\) gives the sample standard \ deviation of the elements in \\!\\(\\*StyleBox[\\\"list\\\", \\\"TI\\\"]\\). \ \\n\\!\\(\\*RowBox[{\\\"StandardDeviation\\\", \\\"[\\\", StyleBox[\\\"dist\\\ \", \\\"TI\\\"], \\\"]\\\"}]\\) gives the standard deviation of the symbolic \ distribution \\!\\(\\*StyleBox[\\\"dist\\\", \\\"TI\\\"]\\).\"\>", "\[NonBreakingSpace]", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/StandardDeviation"]}]], "Print", "PrintUsage", CellChangeTimes->{3.741049246641931*^9}, CellTags-> "Info93741034846-4460184",ExpressionUUID->"0cd2dbe9-8751-48de-942f-\ 3c4bd750ddce"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "N"}]], "Input",ExpressionUUID->"e75c18dd-7ac9-4d1e-b5e1-7876640c6f36"], Cell[BoxData[ RowBox[{"\<\"\\!\\(\\*RowBox[{\\\"N\\\", \\\"[\\\", StyleBox[\\\"expr\\\", \ \\\"TI\\\"], \\\"]\\\"}]\\) gives the numerical value of \\!\\(\\*StyleBox[\\\ \"expr\\\", \\\"TI\\\"]\\). \\n\\!\\(\\*RowBox[{\\\"N\\\", \\\"[\\\", \ RowBox[{StyleBox[\\\"expr\\\", \\\"TI\\\"], \\\",\\\", StyleBox[\\\"n\\\", \\\ \"TI\\\"]}], \\\"]\\\"}]\\) attempts to give a result with \\!\\(\\*StyleBox[\ \\\"n\\\", \\\"TI\\\"]\\)\[Hyphen]digit 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By the computations at \ the end of the simulation, we find, mean = 499.301 standard deviation = 15.922 In part (a) we found (or quoted) the theoretical result =500 and \ \[Sigma]_k=15.811 for a Binomial distribution with p=1/2 and N=1000. \ Therefore the results of the simulation are close to the theoretical results. \ We can therefore consider the distribution of heads for 1000 coin flips to be \ a Gaussian distribution with mean 500 and standard deviation 15.811)\ \>", "Text", CellChangeTimes->{{3.740831895476071*^9, 3.740831896053443*^9}, 3.740844544226406*^9, {3.74105028335826*^9, 3.741050596958383*^9}, 3.741050693865059*^9},ExpressionUUID->"bcc2d23a-e4b3-426c-9f8a-\ 7478971a0a1d"] }, Open ]] }, WindowSize->{954, 655}, WindowMargins->{{4, Automatic}, {Automatic, 4}}, FrontEndVersion->"11.2 for Mac OS X x86 (32-bit, 64-bit Kernel) (September \ 10, 2017)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline 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