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Notebook[{

Cell[CellGroupData[{
Cell["Joint Exponential Probability Distribution", "Subtitle"],

Cell[CellGroupData[{

Cell["Computing Masses and Moments in Probability", "Section"],

Cell[TextData[{
  "In multivariable probability, probability density functions are defined \
over specified regions (domains) in the same way that density functions are \
defined over regions of a solid. These probability functions are designed so \
that the total \"mass\" is always 1. Probabilities are then defined to be the \
integrals of the density functions over a particular subregion of the domain. \
Consider the following example.\n\nSuppose that a national fast-food outlet \
is interested in the joint behavior of the random variables ",
  StyleBox["x",
    FontSlant->"Italic"],
  ", defined as the total time between a customer's arrival at the store and \
departing from the service window, and ",
  StyleBox["y",
    FontSlant->"Italic"],
  ", the time that a customer waits in line before reaching the service \
window. Since ",
  StyleBox["x",
    FontSlant->"Italic"],
  " contains the time a customer waits in line, ",
  StyleBox["x",
    FontSlant->"Italic"],
  " must be greater than ",
  StyleBox["y",
    FontSlant->"Italic"],
  ". The relative frequency distribution of observed values of ",
  StyleBox["x",
    FontSlant->"Italic"],
  " and ",
  StyleBox["y",
    FontSlant->"Italic"],
  " can be modeled by the probability density function:  \n",
  Cell[BoxData[
      FormBox[
        RowBox[{\(f(x, y)\), " ", "=", " ", 
          FormBox[\( .04  e\^\(\(- .1\) x - \(\(.3\)  \(y\)\(\ \)\)\)\),
            "TraditionalForm"]}], TraditionalForm]]],
  "  for  ",
  Cell[BoxData[
      \(TraditionalForm\`0\  \[LessEqual] \ y\  \[LessEqual] \ 
        x\  < \ \[Infinity]\)]],
  "  and 0 elsewhere, where ",
  StyleBox["x",
    FontSlant->"Italic"],
  " and ",
  StyleBox["y",
    FontSlant->"Italic"],
  " are measured in minutes."
}], "Text"],

Cell[BoxData[
    \(f[x_, y_] :=  .04 
         E\^\(\(- .1\) x - \(\(.3\)  \(y\)\(\ \)\)\)\)], "Input"],

Cell["\<\
We examine the domain of this function. First we must load a package to \
assist us in graphing.\
\>", "Text"],

Cell[BoxData[
    \(<< \ Graphics`FilledPlot`\)], "Input"],

Cell[BoxData[
    \(\(FilledPlot[x, {x, 0, 50}, 
        AxesLabel -> {"\<time in system\>", "\<time in line\>"}];\)\)], \
"Input"],

Cell["\<\
We begin by verifying that the integral of the density function over the \
entire domain is 1.\
\>", "Text"],

Cell[BoxData[{
    \(Clear[x, y]\), "\n", 
    \(Integrate[f[x, y], {x, 0, \[Infinity]}, {y, 0, x}]\)}], "Input"],

Cell[TextData[{
  "To compute the average or expected values of ",
  StyleBox["x",
    FontSlant->"Italic"],
  " and ",
  StyleBox["y",
    FontSlant->"Italic"],
  " over the domain, we find the first moments of the density function with \
respect to the",
  StyleBox[" x",
    FontSlant->"Italic"],
  " and ",
  StyleBox["y",
    FontSlant->"Italic"],
  " axes. The second integral is difficult to evaluate as ",
  StyleBox["x",
    FontSlant->"Italic"],
  " approaches \[Infinity], so we substitute a large value of ",
  StyleBox["x",
    FontSlant->"Italic"],
  " for the upper bound."
}], "Text"],

Cell[BoxData[{
    \(Print[\ "\<Expected time in system = \>", 
      Integrate[
        x\ f[x, y], {x, 0, \[Infinity]}, {y, 0, 
          x}], "\< minutes.\>"]\), "\n", 
    \(Print[\ "\<Expected time waiting in line = \>", 
      Integrate[
        y\ f[x, y] // Simplify, {x, 0, 1000000}, {y, 0, 
          x}], "\< minutes.\>"]\)}], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["Computing Probabilities", "Section"],

Cell[TextData[{
  "Now we compute the probability that ",
  StyleBox["y",
    FontSlant->"Italic"],
  " is at least one-half as large as ",
  StyleBox["x",
    FontSlant->"Italic"],
  ". First we can draw the region over which we will integrate (the part for \
",
  StyleBox["x < 5",
    FontSlant->"Italic"],
  "). "
}], "Text"],

Cell[BoxData[
    \(\(FilledPlot[{x, 1/2  x}, {x, 0, 5}, 
        AxesLabel -> {"\<time in system\>", "\<time in line\>"}];\)\)], \
"Input"],

Cell["\<\
The probability of this event is the integral of the density function over \
this region.\
\>", "Text"],

Cell[BoxData[
    \(Print["\<The probability that the time in line is at least half as long \
as the time in the system is \>", 
      Integrate[f[x, y], {x, 0, \[Infinity]}, {y, 1/2  x, x}]]\)], "Input"],

Cell[TextData[{
  "If we want to know the probability that a customer spends no more than six \
minutes at the service window itself, we represent that quantity ",
  Cell[BoxData[
      \(TraditionalForm\`x\  - \ y\)]],
  ", and we integrate over the portion of the domain where ",
  Cell[BoxData[
      \(TraditionalForm\`x\  - \ y\  \[LessEqual] \ 6. \)]]
}], "Text"],

Cell[BoxData[{
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    \(\(p2 = 
        FilledPlot[x, {x, 0, 6}, \ Fills\  -> GrayLevel[ .8], 
          DisplayFunction -> Identity];\)\), "\n", 
    \(\(Show[p1, p2, DisplayFunction -> $DisplayFunction, 
        AxesLabel -> {"\<time in system\>", "\<time in line\>"}];\)\)}], \
"Input"],

Cell[TextData[{
  "Looking at the domain, you can see that it is easier here to integrate \
with respect to ",
  StyleBox["x",
    FontSlant->"Italic"],
  " first."
}], "Text"],

Cell[BoxData[
    \(Print["\<The probability that the time spent at the service desk is no \
more than 6 minutes is \>", 
      Integrate[f[x, y], {y, 0, \[Infinity]}, {x, y, y + 6}]]\)], "Input"]
}, Closed]]
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