------------------------------------- When , n, balls are thrown into, n, boxes The expected value of the random variable : max number of balls in a box is roughly 0.535357422417410462119522258180 + 1.03342541460916489360680729622 ln(n) 2 - 0.0675457450935962804812866011165 ln(n) 3 + 0.00383514347568457694376498908827 ln(n) 4 - 0.0000894290865432026782269212114343 ln(n) and in Maple input form: .535357422417410462119522258180+1.03342541460916489360680729622*ln(n)-.\ 675457450935962804812866011165e-1*ln(n)^2+.383514347568457694376498908827e-2*ln (n)^3-.894290865432026782269212114343e-4*ln(n)^4 Let's see how good it is ---------------------- for n=100, the true value of the average is, 4.23259523867251145796646884086 While our appx. gives:, 4.19631054306703684328485754316 Their ratio is, 0.991427317388171812851326813429 ---------------------- for n=200, the true value of the average is, 4.62986618012845420288620961646 While our appx. gives:, 4.61456455098315697379899977119 Their ratio is, 0.996695016972418695236057678662 ------------------------------------- When , 2 n, balls are thrown into, n, boxes The expected value of the random variable : max number of balls in a box is roughly 1.98908683011881253168578573406 + 1.15743889763939289422418458344 ln(n) 2 - 0.0608782399881914403783567808929 ln(n) 3 + 0.00298688107150454523741931956150 ln(n) 4 - 0.0000626933010363745452646002119698 ln(n) and in Maple input form: 1.98908683011881253168578573406+1.15743889763939289422418458344*ln(n)-.\ 608782399881914403783567808929e-1*ln(n)^2+.298688107150454523741931956150e-2*ln (n)^3-.626933010363745452646002119698e-4*ln(n)^4 Let's see how good it is ---------------------- for n=100, the true value of the average is, 6.30766074530060426889700079995 While our appx. gives:, 6.29172443109790184558922442805 Their ratio is, 0.997473498520893747563498289294 ---------------------- for n=200, the true value of the average is, 6.81759539492974545357941283079 While our appx. gives:, 6.80743058910113356977886610360 Their ratio is, 0.998509033575654626982976387664 ------------------------------------- When , 3 n, balls are thrown into, n, boxes The expected value of the random variable : max number of balls in a box is roughly 3.61669489593082446518298684249 + 1.14054455468735683815890251212 ln(n) 2 - 0.0399388384696551428169159172558 ln(n) 3 + 0.00134012574271136954434408716939 ln(n) 4 - 0.0000191954028841094223013925450498 ln(n) and in Maple input form: 3.61669489593082446518298684249+1.14054455468735683815890251212*ln(n)-.\ 399388384696551428169159172558e-1*ln(n)^2+.134012574271136954434408716939e-2*ln (n)^3-.191954028841094223013925450498e-4*ln(n)^4 Let's see how good it is ---------------------- for n=100, the true value of the average is, 8.12601616787192405093004512394 While our appx. gives:, 8.14433951141072859831978991934 Their ratio is, 1.00225489873023513109070598344 ---------------------- for n=200, the true value of the average is, 8.71521928716572850259045849221 While our appx. gives:, 8.72268924294425499582214865170 Their ratio is, 1.00085711621617220297082789175 ------------------------------------- When , 4 n, balls are thrown into, n, boxes The expected value of the random variable : max number of balls in a box is roughly 4.52084858352817004795118571180 + 1.38810960133027158756571254235 ln(n) 2 - 0.0630108142862176245443051061462 ln(n) 3 + 0.00270959601036440992160107716101 ln(n) 4 - 0.0000514601445124344851190320274620 ln(n) and in Maple input form: 4.52084858352817004795118571180+1.38810960133027158756571254235*ln(n)-.\ 630108142862176245443051061462e-1*ln(n)^2+.270959601036440992160107716101e-2*ln (n)^3-.514601445124344851190320274620e-4*ln(n)^4 Let's see how good it is ---------------------- for n=100, the true value of the average is, 9.81126323882342689895874973867 While our appx. gives:, 9.81850858511145166422675848270 Their ratio is, 1.00073847231611875821313889299 ---------------------- for n=200, the true value of the average is, 14 0.638974518007252251365217785728 10 While our appx. gives:, 10.4691031437294170245943855256 -12 Their ratio is, 0.163842263637978040566861786245 10 Oops, there must be an overflow error, in the computation of the 14 real thing!. The value of, 0.638974518007252251365217785728 10 , for the average does not make sense Let's take instead the Poisson approximation 10.4661236847118827240011341494 Now their ratio is, 1.00028467645780705493342502487 ------------------------------------- When , 5 n, balls are thrown into, n, boxes The expected value of the random variable : max number of balls in a box is roughly 5.65157380707008802241037592666 + 1.51267066248743439649052315718 ln(n) 2 - 0.0685330150790359048795686673552 ln(n) 3 + 0.00288848071780618457183440243708 ln(n) 4 - 0.0000536586025765919655105969747720 ln(n) and in Maple input form: 5.65157380707008802241037592666+1.51267066248743439649052315718*ln(n)-.\ 685330150790359048795686673552e-1*ln(n)^2+.288848071780618457183440243708e-2*ln (n)^3-.536586025765919655105969747720e-4*ln(n)^4 Let's see how good it is ---------------------- for n=100, the true value of the average is, 7058.67302223761722246198732825 While our appx. gives:, 11.4222280256537671416003505592 Their ratio is, 0.00161818347296570084406444806193 Oops, there must be an overflow error, in the computation of the real thing!. The value of, 7058.67302223761722246198732825, for the average does not make sense Let's take instead the Poisson approximation 11.4115078948651997870097115612 Now their ratio is, 1.00093941404478115096672432925 ---------------------- for n=200, the true value of the average is, 24 0.427879907470837318080876461897 10 While our appx. gives:, 12.1296462326795191222965050883 -22 Their ratio is, 0.283482491720091579571631681931 10 Oops, there must be an overflow error, in the computation of the 24 real thing!. The value of, 0.427879907470837318080876461897 10 , for the average does not make sense Let's take instead the Poisson approximation 12.1262045870588330011680871165 Now their ratio is, 1.00028381886483748331738512363 ------------------------------------- When , n, balls are thrown into, 2 n, boxes The expected value of the random variable : max number of balls in a box is roughly 5.97463851296224368330696345520 - 1.54617670359782069010791190785 ln(n) 2 + 0.310925982038018627487209842563 ln(n) 3 - 0.0214193733399047763005862190540 ln(n) 4 + 0.000535280582316640276938088214200 ln(n) and in Maple input form: 5.97463851296224368330696345520-1.54617670359782069010791190785*ln(n)+.31092598\ 2038018627487209842563*ln(n)^2-.214193733399047763005862190540e-1*ln(n)^3+.\ 535280582316640276938088214200e-3*ln(n)^4 Let's see how good it is ---------------------- for n=100, the true value of the average is computed as, 3.28227346472814642285792410925 While our appx. gives:, 3.59705807603608806035512034072 Their ratio is, 1.09590444388338422728915298160 ---------------------- for n=200, the true value of the average is computed as, 3.57229600752345596015012193553 While our appx. gives:, 3.74687988044670554380813682941 Their ratio is, 1.04887161437785842639712877164 ------------------------------------- When , n, balls are thrown into, 3 n, boxes The expected value of the random variable : max number of balls in a box is roughly 0.0991970248216010088577507759874 + 0.799588470346147868003993207101 ln(n) 2 - 0.0578793919452242346730120689591 ln(n) 3 + 0.00327289525600055987149097599774 ln(n) 4 - 0.0000698117015499697329028102796125 ln(n) and in Maple input form: .991970248216010088577507759874e-1+.799588470346147868003993207101*ln(n)-.\ 578793919452242346730120689591e-1*ln(n)^2+.327289525600055987149097599774e-2*ln (n)^3-.698117015499697329028102796125e-4*ln(n)^4 Let's see how good it is ---------------------- for n=100, the true value of the average is computed as, 2.89128554370625404062830911417 While our appx. gives:, 2.84220272122559093700391064500 Their ratio is, 0.983023875802406816338568649158 ---------------------- for n=200, the true value of the average is computed as, 3.17532156352360767147622103755 While our appx. gives:, 3.14265057315373560662728900953 Their ratio is, 0.989710966364736443574665713472 ------------------------------------- When , n, balls are thrown into, 4 n, boxes The expected value of the random variable : max number of balls in a box is roughly -1.82177024876433332308048817711 + 1.39171122812604506591930272421 ln(n) 2 - 0.136363187218728207804616435831 ln(n) 3 + 0.00769415856887460821540710241240 ln(n) 4 - 0.000164202879023876057878438202207 ln(n) and in Maple input form: -1.82177024876433332308048817711+1.39171122812604506591930272421*ln(n)-.1363631\ 87218728207804616435831*ln(n)^2+.769415856887460821540710241240e-2*ln(n)^3-.\ 164202879023876057878438202207e-3*ln(n)^4 Let's see how good it is ---------------------- for n=100, the true value of the average is computed as, 2.63466470673896464837323208400 While our appx. gives:, 2.37295640320107358697031675783 Their ratio is, 0.900667321018689152619817341391 ---------------------- for n=200, the true value of the average is computed as, 2.93414539981233468528119948520 While our appx. gives:, 2.73894044870863789055453754318 Their ratio is, 0.933471275446614905737863792655 ------------------------------------- When , n, balls are thrown into, 5 n, boxes The expected value of the random variable : max number of balls in a box is roughly -11.4605621233248558321707146339 + 5.43252646759285981968699632703 ln(n) 2 - 0.770279892454738507425442899802 ln(n) 3 + 0.0508316139042000539771411282328 ln(n) 4 - 0.00124310205031638561188412144954 ln(n) and in Maple input form: -11.4605621233248558321707146339+5.43252646759285981968699632703*ln(n)-.7702798\ 92454738507425442899802*ln(n)^2+.508316139042000539771411282328e-1*ln(n)^3-.\ 124310205031638561188412144954e-2*ln(n)^4 Let's see how good it is ---------------------- for n=100, the true value of the average is computed as, 2.46377222763347867239466437093 While our appx. gives:, 1.62671257633913126808173708588 Their ratio is, 0.660252826172017395792108016216 Oops, there must be an overflow error, in the computation of the real thing!. The value of, 2.46377222763347867239466437093, for the average does not make sense Let's take instead the Poisson approximation 2.46609639268499594011420991036 Now their ratio is, 0.659630572902312978068069908069 ---------------------- for n=200, the true value of the average is computed as, 2.74312933395878843556830029588 While our appx. gives:, 2.28009197793705825319402542960 Their ratio is, 0.831201048273764458511913536231 This took, 58.555, seconds .