Forbes 400 Data

<< Graphics`Graphics`

<< Statistics`

\(\(wealthlist\  = 
{{1, "Gates, William Henry III", 46}, {2, "Buffett, Warren Edward", 
            36}, {3, "Allen, Paul Gardner", 
            22}, {4, "Walton, Alice L", 20.5}, {4, "Walton, Helen R", 
            20.5}, {4, "Walton, Jim C", 20.5}, {4, "Walton, John T", 
            20.5}, {4, "Walton, S Robson", 
            20.5}, {9, "Ellison, Lawrence Joseph", 
            18.0}, {10, "Dell, Michael", 
            13.0}, {11, "Ballmer, Steven Anthony", 
            12.2}, {12, "Anthony, Barbara Cox", 
            11.0}, {12, "Chambers, Anne Cox", 
            11.0}, {14, "Kluge, John Werner", 
            10.5}, {15, "Mars, Forrest Edward Jr", 
            10.4}, {15, "Mars, Jacqueline", 
            10.4}, {15, "Mars, John Franklyn", 
            10.4}, {18, "Johnson, Abigail", 
            9.8}, {19, "Redstone, Sumner M", 
            9.7}, {20, "Ergen, Charles", 
            8.9}, {21, "Newhouse, Donald Edward", 
            7.7}, {21, "Newhouse, Samuel Irving Jr", 
            7.7}, {23, "Pritzker, Robert Alan", 
            7.6}, {23, "Pritzker, Thomas J", 
            7.6}, {25, "Johnson, Samuel Curtis", 
            7.4}, {26, "Icahn, Carl", 
            7.3}, {27, "Murdoch, Keith Rupert", 
            7.2}, {28, "Soros, George", 
            7.0}, {29, "Omidyar, Pierre M", 
            6.9}, {30, "Warner, H Ty", 6.0}, {31, "Knight, Philip H", 
            5.8}, {32, "Bezos, Jeffrey P", 
            5.1}, {33, "Anschutz, Philip F", 
            5.0}, {33, "Kerkorian, Kirk", 
            5.0}, {33, "Moore, Gordon Earle", 
            5.0}, {36, "Bloomberg, Michael Rubens", 
            4.9}, {36, "Johnson, Edward Crosby III", 
            4.9}, {38, "Davis, Marvin H", 4.7}, {39, "Arison, Micky", 
            4.4}, {40, "Bren, Donald L", 4.0}, {40, "Geffen, David", 
            4.0}, {40, "Koch, Charles De Ganahl", 
            4.0}, {40, "Koch, David Hamilton", 
            4.0}, {44, "Abele, John E", 3.9}, {45, "Broad, Eli", 
            3.8}, {46, "Perot, Henry Ross", 
            3.7}, {47, "Menard, John R Jr", 
            3.5}, {47, "Perelman, Ronald Owen", 
            3.5}, {49, "Nicholas, Peter M", 
            3.3}, {49, "Wrigley, William Jr", 
            3.3}, {51, "Bechtel, Riley P", 
            3.2}, {51, "Bechtel, Stephen Davison Jr", 
            3.2}, {51, "Crown, Lester & family", 
            3.2}, {51, "Schwab, Charles R", 
            3.2}, {55, "Cook, William Alfred", 
            3.1}, {56, "Greenberg, Maurice Raymond", 
            3.0}, {56, "Kaiser, George B", 
            3.0}, {56, "Kroenke, Ann Walton", 
            3.0}, {56, "Lucas, George", 
            3.0}, {56, "Taylor, Jack Crawford", 
            3.0}, {61, "Laurie, Nancy Walton", 
            2.9}, {62, "Goodnight, James", 
            2.8}, {62, "Hillman, Henry Lea", 
            2.8}, {62, "Perenchio, A Jerrold", 
            2.8}, {62, "Schulze, Richard M", 
            2.8}, {66, "Sorenson, James L", 
            2.7}, {66, "Wexner, Leslie Herbert", 
            2.7}, {68, "Bass, Robert Muse", 
            2.6}, {68, "Buffett, Susan Thompson", 
            2.6}, {68, "Lauder, Leonard Alan", 
            2.6}, {71, "Huntsman, Jon Meade", 
            2.5}, {71, "Rockefeller, David Sr", 
            2.5}, {71, "Rowling, Robert", 
            2.5}, {71, "Spielberg, Steven Allen", 
            2.5}, {71, "Trump, Donald John", 
            2.5}, {76, "Bronfman, Edgar M Sr", 
            2.4}, {76, "Simplot, John Richard & family", 
            2.4}, {78, "Butt, Charles C", 2.3}, {78, "Hunt, Ray Lee", 
            2.3}, {78, "Ingram, Martha R & family", 
            2.3}, {78, "Jobs, Steven Paul", 
            2.3}, {78, "Lauren, Ralph", 
            2.3}, {78, "Tisch, Preston Robert", 
            2.3}, {78, "Turner, Robert E " Ted "", 
            2.3}, {85, "Helmsley, Leona", 
            2.2}, {85, "Johnson, Barbara Piasecka", 
            2.2}, {85, "Stern, Leonard Norman", 
            2.2}, {88, "Fisher, Donald George", 
            2.1}, {88, "Fisher, Doris Feigenbaum", 
            2.1}, {88, "Getty, Gordon Peter", 
            2.1}, {88, "Hostetter, Amos Barr Jr", 
            2.1}, {88, "Pohlad, Carl", 
            2.1}, {88, "Spangler, Clemmie Dixon Jr", 
            2.1}, {88, "Zell, Samuel", 
            2.1}, {95, "Annenberg, Leonore", 
            2.0}, {95, "Johnson, Charles Bartlett", 
            2.0}, {95, "Kellogg, Peter R", 
            2.0}, {95, "Marcus, Bernard", 
            2.0}, {95, "McCaw, Craig O", 
            2.0}, {95, "McGovern, Patrick Joseph", 
            2.0}, {95, "Rainwater, Richard Edward", 
            2.0}, {95, "Tisch, Laurence Alan", 
            2.0}, {95, "Udvar-Hazy, Steven", 
            2.0}, {104, "Davidson, William Morse", 
            1.9}, {104, "Hall, Donald Joyce", 
            1.9}, {104, "Kim, James & family", 
            1.9}, {104, "Lerner, Randolph D", 
            1.9}, {104, "Malone, John C", 
            1.9}, {104, "Rich, Robert Edward Sr", 
            1.9}, {104, "Taylor, Glen", 
            1.9}, {111, "Abraham, S Daniel", 
            1.8}, {111, "Adelson, Sheldon", 
            1.8}, {111, "Allen, Herbert Anthony Jr", 
            1.8}, {111, "Burkle, Ronald", 
            1.8}, {111, "Hughes, Bradley Wayne", 
            1.8}, {111, "Huizenga, H Wayne", 
            1.8}, {111, "Jackson, Jess Stonestreet", 
            1.8}, {111, "Kovner, Bruce", 
            1.8}, {111, "Lauder, Ronald Steven", 
            1.8}, {111, "Washington, Dennis", 
            1.8}, {121, "DeVos, Richard M", 
            1.7}, {121, "Kroc, Joan Beverly", 
            1.7}, {121, "Mathile, Clayton Lee", 
            1.7}, {121, "Rales, Steven", 1.7}, {121, "Saban, Haim", 
            1.7}, {126, "Diller, Barry", 
            1.6}, {126, "Druckenmiller, Stanley", 
            1.6}, {126, "Filo, David", 
            1.6}, {126, "Golisano, Blase Thomas", 
            1.6}, {126, "Gonda, Louis L", 1.6}, {126, "Gores, Alec", 
            1.6}, {126, "Gores, Tom T", 
            1.6}, {126, "Johnson, Rupert Harris Jr", 
            1.6}, {126, "Marshall, E Pierce", 
            1.6}, {126, "Mitchell, George Phydias", 
            1.6}, {126, "Morgridge, John P", 
            1.6}, {126, "Nicholas, Henry T III", 
            1.6}, {126, "Rales, Mitchell", 
            1.6}, {126, "Stempel, Ernest E", 
            1.6}, {140, "Batten, Frank Sr", 
            1.5}, {140, "Booth, Franklin Otis Jr", 
            1.5}, {140, "Cargill, James R", 
            1.5}, {140, "Cargill, Margaret Anne", 
            1.5}, {140, "Farmer, Richard T", 
            1.5}, {140, "Frist, Thomas F Jr & family", 
            1.5}, {140, "Kohler, Herbert & family", 
            1.5}, {140, "Krasny, Michael", 
            1.5}, {140, "Kroenke, E Stanley", 
            1.5}, {140, "Lampert, Edward S", 
            1.5}, {140, "Malone, Mary Alice Dorrance", 
            1.5}, {140, "Mays, L Lowry", 
            1.5}, {140, "Murdock, David Howard", 
            1.5}, {140, "Rockefeller, Laurance Spelman", 
            1.5}, {140, "Sarofim, Fayez Shalaby", 
            1.5}, {140, "Smith, Frederick Wallace", 
            1.5}, {140, "Sperling, John G", 
            1.5}, {140, "Sperling, Peter", 
            1.5}, {140, "Stephens, Jackson Thomas", 
            1.5}, {140, "Stryker, Jon Lloyd", 
            1.5}, {140, "Stryker, Ronda E", 
            1.5}, {140, "Van Andel, Jay", 
            1.5}, {162, "Bass, Lee Marshall", 
            1.4}, {162, "Dolan, Charles Francis", 
            1.4}, {162, "Dorrance, Bennett", 
            1.4}, {162, "Gage, Barbara Carlson & family", 
            1.4}, {162, "Kinder, Richard", 
            1.4}, {162, "Moran, James Martin", 
            1.4}, {162, "Munger, Charles T", 
            1.4}, {162, "Naify, Robert Allen", 
            1.4}, {162, "Nelson, Marilyn Carlson & family", 
            1.4}, {162, "Ricketts, J Joeseph & family", 
            1.4}, {162, "Sall, John", 1.4}, {162, "Samueli, Henry", 
            1.4}, {162, "Simmons, Harold Clark", 
            1.4}, {162, "Waitt, Theodore W", 
            1.4}, {162, "Weill, Sanford", 
            1.4}, {162, "Williams, Arthur L Jr", 
            1.4}, {162, "Yang, Jerry", 
            1.4}, {179, "Beck, Nancy Lerner", 
            1.3}, {179, "Cuban, Mark", 1.3}, {179, "Egan, Richard J", 
            1.3}, {179, "Ford, Gerald J", 
            1.3}, {179, "Gonda, Leslie L", 
            1.3}, {179, "Hearst, William Randolph III", 
            1.3}, {179, "Johnson, Robert L", 
            1.3}, {179, "Kellogg, William S", 
            1.3}, {179, "Kravis, Henry R", 
            1.3}, {179, "Lerner, Norma", 
            1.3}, {179, "Lewis, Peter Benjamin", 
            1.3}, {179, "Lindner, Carl Henry Jr", 
            1.3}, {179, "McNair, Robert C", 
            1.3}, {179, "Roberts, George R", 
            1.3}, {179, "Sobrato, John Albert", 
            1.3}, {179, "Stryker, Pat A", 
            1.3}, {195, "Berg, Carl Edwin", 
            1.2}, {195, "Cooke, Phoebe Hearst", 
            1.2}, {195, "Duffield, David A", 
            1.2}, {195, "Emmerson, Archie Aldis " Red "", 
            1.2}, {195, "Field, Frederick Woodruff", 
            1.2}, {195, "Flatley, Thomas John", 
            1.2}, {195, "France, James C", 
            1.2}, {195, "France, William C Jr", 
            1.2}, {195, "Fung, Victor", 
            1.2}, {195, "Gates, Charles Cassius Jr", 
            1.2}, {195, "Goldsbury, Christopher", 
            1.2}, {195, "Hearst, Austin", 
            1.2}, {195, "Hearst, David Whitmire Jr", 
            1.2}, {195, "Hearst, George Randolph Jr", 
            1.2}, {195, "Holding, Robert Earl", 
            1.2}, {195, "Hubbard, Stanley Stub", 
            1.2}, {195, "Jamail, Joseph Dahr Jr", 
            1.2}, {195, "Levine, Leon & family", 
            1.2}, {195, "Marriott, Richard Edwin", 
            1.2}, {195, "McLane, Robert Drayton Jr", 
            1.2}, {195, "Morean, William", 
            1.2}, {195, "Rockefeller, Winthrop Paul", 
            1.2}, {195, "Siebel, Thomas M", 
            1.2}, {195, "Soon-Shiong, Patrick", 
            1.2}, {195, "Ueltschi, Albert Lee", 
            1.2}, {195, "Ziff, Daniel Morton", 
            1.2}, {195, "Ziff, Dirk Edward", 
            1.2}, {195, "Ziff, Robert David", 
            1.2}, {195, "Zuckerman, Mortimer Benjamin", 
            1.2}, {224, "Arrillaga, John", 
            1.1}, {224, "Bass, Sid Richardson", 
            1.1}, {224, "Blank, Arthur M", 1.1}, {224, "Gerry, Alan", 
            1.1}, {224, "Green, David", 1.1}, {224, "Green, Pincus", 
            1.1}, {224, "Jannard, James", 
            1.1}, {224, "Lindemann, George L & family", 
            1.1}, {224, "Magness, Gary", 
            1.1}, {224, "McCaw, John Elroy Jr", 
            1.1}, {224, "McCombs, Billy Joe " Red "", 
            1.1}, {224, "Peery, Richard Taylor", 
            1.1}, {224, "Rich, Marc David", 
            1.1}, {224, "Scaife, Richard Mellon", 
            1.1}, {224, "Smith, Ollen Bruton", 
            1.1}, {224, "Troutt, Kenny A", 
            1.1}, {224, "Weber, Charlotte Colket", 
            1.1}, {224, "Whitman, Margaret", 
            1.1}, {224, "Winfrey, Oprah", 
            1.1}, {224, "Zachry, Henry Bartell Jr", 
            1.1}, {244, "Anderson, John Edward", 
            1.0}, {244, "Bluhm, Neil Gary", 
            1.0}, {244, "Cohen, Steven A", 
            1.0}, {244, "Comer, Gary Campbell", 
            1.0}, {244, "Cook, Scott D", 
            1.0}, {244, "Glazer, Malcolm", 
            1.0}, {244, "Jacobs, Irwin Mark", 
            1.0}, {244, "Kohlberg, Jerome Spiegel Jr", 
            1.0}, {244, "Mann, Alfred", 
            1.0}, {244, "Marriott, John Willard Jr", 
            1.0}, {244, "McCaw, Bruce R", 
            1.0}, {244, "Milliken, Roger", 
            1.0}, {244, "Moreno, Arturo", 
            1.0}, {244, "Rock, Arthur J", 
            1.0}, {244, "Scott, Walter Jr", 
            1.0}, {244, "Siegel, Herbert", 
            1.0}, {244, "Simon, Melvin", 
            1.0}, {244, "Simonyi, Charles", 
            1.0}, {244, "White, Dean V", 
            1.0}, {263, "Debartolo, Edward John Jr",  .990}, {263, "Roski, Edward P Jr",  .990}, {265, "Argyros, George Leon",  .980}, {266, "Bailey, Thomas",  .975}, {266, "Keinath, Pauline MacMillan",  .975}, {266, "MacMillan, Cargill Jr",  .975}, {266, "MacMillan, John Hugh III",  .975}, {266, "MacMillan, W Duncan",  .975}, {266, "MacMillan, Whitney",  .975}, {266, "Opperman, Dwight D",  .975}, {266, "Pictet, Marion MacMillan",  .975}, {266, "West, Alfred P Jr",  .975}, {275, "Herb, Marvin J",  .970}, {275, "Kao, Min H",  .970}, {275, "Wyly, Samuel",  .970}, {278, "Copley, Helen K",  .960}, {278, "Edson, John Orin",  .960}, {278, "Harbert, Marguerite",  .960}, {278, "Marshall, Barbara Hall",  .960}, {278, "Reid, Elizabeth Ann Hall",  .960}, {283, "Ansin, Edmund Newton",  .950}, {283, "Ford, William Clay Sr",  .950}, {283, "Jones, Jerral W",  .950}, {283, "Peltz, Nelson",  .950}, {287, "Joseph, George",  .940}, {287, "Levine, William S",  .940}, {289, "Chace, Malcolm III & family",  .935}, {290, "Gallo, Ernest",  .930}, {290, "Robinson, Jesse Mack",  .930}, {292, "Pasculano, Lynne",  .910}, {292, "Van Beuren, Hope Hill",  .910}, {294, "Bowes, William",  .900}, {294, "Connor, William E II",  .900}, {294, "Disney, Roy Edward",  .900}, {294, "Krehbiel, John Hammond Jr",  .900}, {294, "Lee, Thomas Haskell",  .900}, {294, "Magerko, Maggie Hardy",  .900}, {294, "Price, Michael F",  .900}, {294, "Teel, Joyce Raley",  .900}, {294, "Watkins, Edward G III",  .900}, {303, "Jacobs, Jeremy Maurice Sr",  .875}, {304, "Dedman, Nancy & family",  .870}, {305, "Anselmo, Mary",  .860}, {305, "Gold, David & family",  .860}, {305, "Spanos, Alexander Gus",  .860}, {308, "Heisley, Michael E Sr",  .850}, {308, "Honickman, Harold",  .850}, {310, "Glazer, Guilford",  .840}, {311, "Lenfest, Harold FitzGerald",  .825}, {311, "Milken, Michael Robert",  .825}, {311, "Oates, Marvin L " 
              Buzz "",  .825}, {314, "Langone, Kenneth G",  .820}, {315, "Bass, Edward Perry",  .810}, {315, "Burrell, Gary L",  .810}, {315, "Taubman, A Alfred",  .810}, {318, "Bose, Amar Gopal",  .800}, {318, "Boyle, Timothy",  .800}, {318, "Connell, Grover",  .800}, {318, "Katzenberg, Jeffrey",  .800}, {318, "Marion, Anne Windfohr",  .800}, {318, "Marsico, Thomas",  .800}, {318, "Strong, Richard S",  .800}, {325, "Saul, Bernard Francis II",  .790}, {326, "Fireman, Paul Barry",  .780}, {326, "Robertson, Julian H Jr",  .780}, {328, "Abramson, Leonard",  .775}, {328, "Casden, Alan I",  .775}, {328, "Earhart, Anne Catherine Getty",  .775}, {328, "Feld, Kenneth",  .775}, {328, "Getty, Caroline Marie",  .775}, {328, "Hamilton, Dorrance Hill",  .775}, {328, "Lupton, John Thomas",  .775}, {328, "Perry, Claire Eugenia Getty",  .775}, {328, "Speer, Roy Merrill",  .775}, {337, "Herma, John",  .770}, {337, "Rizzuto, Leandro P",  .770}, {339, "Hascoe, Norman",  .760}, {340, "May, Cordelia Scaife",  .750}, {340, "Osher, Bernard A",  .750}, {340, "Ruffin, Phillip",  .750}, {340, "Schneider, Donald J",  .750}, {340, "Shorenstein, Walter Herbert",  .750}, {340, "Vilar, Alberto",  .750}, {340, "Wagner, Todd R",  .750}, {347, "Brown, John W",  .740}, {347, "Dehaan, Christel",  .740}, {347, "Moores, John Jay",  .740}, {350, "Forstmann, Theodore",  .725}, {350, "Hicks, Thomas O",  .725}, {350, "Thorne, Oakleigh B",  .725}, {353, "Bass, Perry Richardson",  .720}, {353, "Butler, Sarah Turner",  .720}, {353, "Corn, Elizabeth Turner",  .720}, {353, "Turner, William Bradley",  .720}, {357, "Clark, James H",  .700}, {357, "Congel, Robert J",  .700}, {357, "Day, Robert Addison",  .700}, {357, "Khosla, Vinod",  .700}, {357, "Kimmel, Sidney",  .700}, {357, "Petersen, Robert Einar",  .700}, {357, "Pulte, William J",  .700}, {357, "Solow, Sheldon Henry",  .700}, {357, "Taylor, Patrick F",  .700}, {357, "Waitt, Norman W Jr",  .700}, {357, "Wendt, Richard",  .700}, {368, "Fisher, Max Martin",  .680}, {368, "Sandler, Herbert M",  .680}, {368, "Sandler, Marion O",  .680}, {368, "Strawbridge, George Jr",  .680}, {372, "Johnson, Bobby Ray Jr",  .675}, {372, "Neubauer, Joseph",  .675}, {374, "Krehbiel, Frederick A",  .670}, {375, "Manoogian, Richard Alexander",  .660}, {375, "Ryan, Patrick George",  .660}, {377, "Friedkin, Thomas",  .650}, {377, "Griffin, Ken",  .650}, {377, "Jones, Paul Tudor II",  .650}, {377, "Koch, Frederick Robinson",  .650}, {377, "Koch, William Ingraham",  .650}, {377, "Nash, Jack",  .650}, {377, "Wynn, Stephen A",  .650}, {384, "McEvoy, Nan Tucker",  .640}, {385, "Eisner, Michael D",  .630}, {385, "Green, Joshua III",  .630}, {385, "Tyson, Donald John",  .630}, {388, "Bisciotti, Stephen J",  .625}, {388, "Levine, Stuart Robert",  .625}, {388, "McKelvey, Andrew",  .625}, {388, "Milken, Lowell Jay",  .625}, {388, "Roberts, Brian L",  .625}, {393, "Baker, Jay H",  .610}, {393, "Case, Stephen M",  .610}, {393, "Smith, Mark C",  .610}, {396, "Carsey, Marcy",  .600}, {396, "Chernick, Aubrey",  .600}, {396, "Gerstner, Louis V Jr",  .600}, {396, "Haworth, Richard G",  .600}, {396, "Werner, Tom",  .600}
};\)
\)
wealths  = {};
\(\(Do[
        AppendTo[wealths, wealthlist[\([i, 3]\)]], {i, 1, 
          Length[wealthlist]}];\)
\)
NN  =  Length[wealths];
\(\(Print["N = ", NN];\)
\)
ave  =  Mean[wealths];
\(\(Print["ave = ", N[ave]];\)
\)
min  =  Min[wealths]; 
Print["min = ", min];
max  =  Max[wealths]; 
\(\(Print["max = ", max];\)
\)
var  =  Variance[wealths]; 
Print["var = ", N[var]]; 
Print["std. dev. = ", 
        N[Sqrt[var]]];

hist  =  
Histogram[
wealths, 
DisplayFunction ->  
            Identity
];
\(\(curve\  = \ 
Plot[
2.5\ NN\ 2\ PDF[NormalDistribution[0, Sqrt[var]], x], {x, 0, max}, 
          DisplayFunction -> \ 
            Identity, 
PlotStyle -> 
            RGBColor[0, 0, 
              1]
];\)
\)
Show[
hist, 
curve, 
DisplayFunction ->  $DisplayFunction
];

Probability of values greater than 30

N[
      Integrate[
        2 PDF[NormalDistribution[0, Sqrt[var]], x], {x, 30, 
          Infinity}]]

Plot of ranked Forbes 400 data

rankwealth  = {};
Do[
AppendTo[
          rankwealth, {wealthlist[[i, 1]], wealthlist[[i, 3]]}], {i, 
          1, Length[
            wealthlist]}
];

    
plot1  =  
ListPlot[
rankwealth, 
PlotStyle ->  
            RGBColor[1, 0, 
              0], 
Prolog -> {PointSize[0.015], 
              Thickness[0.005]}, 
PlotRange ->  
            All
];

Plot of ranked sample from positive normal distribution

NDarray  =  
        Sort[Abs[RandomArray[NormalDistribution[0, Sqrt[var]], 400]], 
          Greater];
rankND  =  
        Table[{i, NDarray[[i]]}, {i, 1, 
            400}];
plot3  =  
ListPlot[
rankND, 
PlotStyle ->  
            RGBColor[1, 0, 
              0], 
Prolog -> {PointSize[0.015], 
              Thickness[
                0.005]}, 
PlotRange ->  {All, {0, 
                47}}
];

Log-log plot of ranked Forbes 400 data

plot2  =  
ListPlot[
Log[
            rankwealth], 
PlotStyle ->  
            RGBColor[1, 0, 
              0], 
Prolog -> {PointSize[0.015], 
              Thickness[0.005]}, 
PlotRange ->  
            All
];

Linear fit to ranked Forbes 400 data

fit  =  
        Fit[Log[rankwealth], {1, logr}, logr];
Print[fit];

    
line  = 
 
        Plot[
fit, {logr, 0, 
            Log[400]}, 
PlotStyle -> 
            RGBColor[0, 0, 1], 
DisplayFunction ->  
            Identity
];

    
Show[
plot2, 
        line, 
DisplayFunction ->  $DisplayFunction
];

Log-log plot of ranked sample from positive normal distribution

plot4  =  
ListPlot[
Log[
            rankND], 
PlotStyle ->  
            RGBColor[1, 0, 
              0], 
Prolog -> {PointSize[0.015], 
              Thickness[0.005]}, 
PlotRange ->  
            All
];