(* Content-type: application/vnd.wolfram.mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='Mathematica 11.3' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       158,          7]
NotebookDataLength[      7724,        231]
NotebookOptionsPosition[      6373,        197]
NotebookOutlinePosition[      6763,        214]
CellTagsIndexPosition[      6720,        211]
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Notebook[{

Cell[CellGroupData[{
Cell[BoxData[
 RowBox[{"PDF", "[", 
  RowBox[{
   RowBox[{"Min", "[", 
    RowBox[{"x", ",", "y"}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"x", "\[Distributed]", 
      RowBox[{"ExponentialDistribution", "[", "2", "]"}]}], ",", 
     RowBox[{"y", "\[Distributed]", 
      RowBox[{"ExponentialDistribution", "[", "2", "]"}]}]}], "}"}]}], 
  "]"}]], "Input",
 CellChangeTimes->{{3.803012990056808*^9, 3.803012998671767*^9}, {
  3.8030130489929147`*^9, 3.8030130592709827`*^9}},
 CellLabel->"In[6]:=",ExpressionUUID->"8a04cf81-6249-419b-bd28-45482dfc51f9"],

Cell[BoxData[
 RowBox[{"PDF", "[", 
  RowBox[{
   RowBox[{"Min", "[", 
    RowBox[{"x", ",", "y"}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"x", "\[Distributed]", 
      RowBox[{"ExponentialDistribution", "[", "2", "]"}]}], ",", 
     RowBox[{"y", "\[Distributed]", 
      RowBox[{"ExponentialDistribution", "[", "2", "]"}]}]}], "}"}]}], 
  "]"}]], "Output",
 CellChangeTimes->{{3.8030129754638047`*^9, 3.803012999377796*^9}, 
   3.8030130595939445`*^9},
 CellLabel->"Out[6]=",ExpressionUUID->"4d57f1b9-807a-4e85-ad08-06b6459fe175"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"Probability", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"Min", "[", 
      RowBox[{"x", ",", "y"}], "]"}], "<", "3"}], ",", 
    RowBox[{"{", 
     RowBox[{
      RowBox[{"x", "\[Distributed]", 
       RowBox[{"ExponentialDistribution", "[", "2", "]"}]}], ",", 
      RowBox[{"y", "\[Distributed]", 
       RowBox[{"ExponentialDistribution", "[", "2", "]"}]}]}], "}"}]}], "]"}],
   "//", "FullSimplify"}]], "Input",
 CellChangeTimes->{{3.8030130027667665`*^9, 3.8030130447979774`*^9}, {
  3.803013171328641*^9, 3.803013184581671*^9}},
 CellLabel->"In[10]:=",ExpressionUUID->"51f9ebaa-7986-4677-b9e8-099b60ce2ab8"],

Cell[BoxData[
 RowBox[{"1", "-", 
  FractionBox["1", 
   SuperscriptBox["\[ExponentialE]", "12"]]}]], "Output",
 CellChangeTimes->{{3.8030130126317663`*^9, 3.803013045117915*^9}, {
  3.8030131812786407`*^9, 3.8030131847736425`*^9}},
 CellLabel->"Out[10]=",ExpressionUUID->"ab8c0c48-b5bd-48fc-a39b-2b6c036777d9"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Probability", "[", 
  RowBox[{
   RowBox[{
    RowBox[{"x", "+", "y"}], "<", "1"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"x", "\[Distributed]", 
      RowBox[{"BinomialDistribution", "[", 
       RowBox[{"3", ",", "p"}], "]"}]}], ",", 
     RowBox[{"y", "\[Distributed]", 
      RowBox[{"BinomialDistribution", "[", 
       RowBox[{"3", ",", "p"}], "]"}]}]}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.803013077384956*^9, 3.803013101779068*^9}},
 CellLabel->"In[8]:=",ExpressionUUID->"87491a1a-9fd8-4856-9fc3-ff867a5d33f7"],

Cell[BoxData[
 SuperscriptBox[
  RowBox[{"(", 
   RowBox[{
    RowBox[{"-", "1"}], "+", "p"}], ")"}], "6"]], "Output",
 CellChangeTimes->{{3.803013099223008*^9, 3.8030131019180465`*^9}},
 CellLabel->"Out[8]=",ExpressionUUID->"8cc2a5c9-e8c2-4c96-80e8-67bddbd7ff75"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Probability", "[", 
  RowBox[{
   RowBox[{
    RowBox[{"x", "-", "y"}], "\[GreaterEqual]", "1"}], ",", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"x", ",", "y"}], "}"}], "\[Distributed]", "  ", 
    RowBox[{"MultinormalDistribution", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{"1", ",", "0"}], "}"}], ",", 
      RowBox[{"{", 
       RowBox[{
        RowBox[{"{", 
         RowBox[{"4", ",", 
          RowBox[{
           RowBox[{
            RowBox[{"-", "1"}], "/", "2"}], " ", "*", "2"}]}], "}"}], ",", 
        RowBox[{"{", 
         RowBox[{
          RowBox[{"-", "1"}], ",", "1"}], "}"}]}], "}"}]}], "]"}]}]}], 
  "]"}]], "Input",
 CellChangeTimes->{{3.803013651407669*^9, 3.803013653367703*^9}, {
  3.8030137578219223`*^9, 3.803013765356889*^9}, {3.8030138007794337`*^9, 
  3.8030138689322824`*^9}},
 CellLabel->"In[12]:=",ExpressionUUID->"3161d5c4-440c-4f93-b30e-34c42fcba097"],

Cell[BoxData[
 FractionBox["1", "2"]], "Output",
 CellChangeTimes->{{3.803013848339901*^9, 3.803013869198244*^9}},
 CellLabel->"Out[12]=",ExpressionUUID->"c6211cb2-b699-45ff-95ba-b4f7a540eab0"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"\:4e8c\:7ef4\:6b63\:6001", "[", 
   RowBox[{
   "\[Micro]1_", ",", "\[Micro]2_", ",", "\[Sigma]12_", ",", "\[Sigma]22_", 
    ",", "\[Rho]_"}], "]"}], "=", 
  RowBox[{"MultinormalDistribution", "[", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"\[Micro]1", ",", "\[Micro]2"}], "}"}], ",", 
    RowBox[{"{", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{"\[Sigma]12", ",", 
        RowBox[{"\[Rho]", " ", 
         RowBox[{"Sqrt", "[", 
          RowBox[{"\[Sigma]12", " ", "\[Sigma]22"}], "]"}]}]}], "}"}], ",", 
      RowBox[{"{", 
       RowBox[{
        RowBox[{"\[Rho]", " ", 
         RowBox[{"Sqrt", "[", 
          RowBox[{"\[Sigma]12", " ", "\[Sigma]22"}], "]"}]}], ",", 
        "\[Sigma]22"}], "}"}]}], "}"}]}], "]"}]}]], "Input",
 CellChangeTimes->{{3.803013903305318*^9, 3.8030140235706673`*^9}},
 CellLabel->"In[13]:=",ExpressionUUID->"e25a9c5f-b609-48f2-9a24-1e041eeaa72c"],

Cell[BoxData[
 RowBox[{"MultinormalDistribution", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"\[Micro]1", ",", "\[Micro]2"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"\[Sigma]12", ",", 
       RowBox[{"\[Rho]", " ", 
        SqrtBox[
         RowBox[{"\[Sigma]12", " ", "\[Sigma]22"}]]}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{"\[Rho]", " ", 
        SqrtBox[
         RowBox[{"\[Sigma]12", " ", "\[Sigma]22"}]]}], ",", "\[Sigma]22"}], 
      "}"}]}], "}"}]}], "]"}]], "Output",
 CellChangeTimes->{3.803014023898626*^9},
 CellLabel->"Out[13]=",ExpressionUUID->"14644ae3-0bfb-49e9-9da7-89cefe506b28"]
}, Open  ]]
},
WindowSize->{1778, 792},
WindowMargins->{{Automatic, -54}, {-296, Automatic}},
Magnification:>1.8 Inherited,
FrontEndVersion->"11.3 for Microsoft Windows (64-bit) (2018\:5e743\:670828\
\:65e5)",
StyleDefinitions->"Default.nb"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[CellGroupData[{
Cell[580, 22, 570, 14, 78, "Input",ExpressionUUID->"8a04cf81-6249-419b-bd28-45482dfc51f9"],
Cell[1153, 38, 548, 14, 58, "Output",ExpressionUUID->"4d57f1b9-807a-4e85-ad08-06b6459fe175"]
}, Open  ]],
Cell[CellGroupData[{
Cell[1738, 57, 650, 16, 78, "Input",ExpressionUUID->"51f9ebaa-7986-4677-b9e8-099b60ce2ab8"],
Cell[2391, 75, 311, 6, 83, "Output",ExpressionUUID->"ab8c0c48-b5bd-48fc-a39b-2b6c036777d9"]
}, Open  ]],
Cell[CellGroupData[{
Cell[2739, 86, 565, 14, 78, "Input",ExpressionUUID->"87491a1a-9fd8-4856-9fc3-ff867a5d33f7"],
Cell[3307, 102, 264, 6, 58, "Output",ExpressionUUID->"8cc2a5c9-e8c2-4c96-80e8-67bddbd7ff75"]
}, Open  ]],
Cell[CellGroupData[{
Cell[3608, 113, 924, 26, 78, "Input",ExpressionUUID->"3161d5c4-440c-4f93-b30e-34c42fcba097"],
Cell[4535, 141, 193, 3, 79, "Output",ExpressionUUID->"c6211cb2-b699-45ff-95ba-b4f7a540eab0"]
}, Open  ]],
Cell[CellGroupData[{
Cell[4765, 149, 925, 24, 78, "Input",ExpressionUUID->"e25a9c5f-b609-48f2-9a24-1e041eeaa72c"],
Cell[5693, 175, 664, 19, 100, "Output",ExpressionUUID->"14644ae3-0bfb-49e9-9da7-89cefe506b28"]
}, Open  ]]
}
]
*)

(* End of internal cache information *)