Statistical Analysis of the Number of Donors a Beggar Needs Before He (or She) can go home By Shalosh B. Ekhad In a certain country there are only, 4, types of coins, of denominations, 1, 5, 10, 25 A person passing a beggar throws A coin of, 1, cents , with probability, p1 A coin of, 5, cents , with probability, p5 A coin of, 10, cents , with probability, p10 A coin of, 25, cents , with probability, 1 - p1 - p5 - p10 Each person's donation is independent of the others. The beggar decides to go home as soon as the amount in his hat exceeds n cen\ ts. Let X(n) be the random variable "Number of donors until he goes home". Since the expected size of a single donation is -24 p1 - 20 p5 - 15 p10 + 25 n The expectation of X(n) is roughly, ---------------------------- -24 p1 - 20 p5 - 15 p10 + 25 More precisely, up to exponentially small error, the Expectation of X(n) is 5 (-60 + 60 p1 + 58 p5 + 51 p10) n - -------------------------------- - ---------------------------- 2 -25 + 24 p1 + 20 p5 + 15 p10 (-25 + 24 p1 + 20 p5 + 15 p10) and in Maple notation it is -5*(-60+60*p1+58*p5+51*p10)/(-25+24*p1+20*p5+15*p10)^2-1/(-25+24*p1+20*p5+15* p10)*n The variance of X(n), up to exponentially small error is 3 2 2 2 5 (34560 p1 + 72576 p1 p10 + 91008 p1 p5 + 50220 p1 p10 + 126720 p1 p10 p5 2 3 2 2 3 + 79680 p1 p5 + 11475 p10 + 43650 p10 p5 + 55200 p10 p5 + 23200 p5 2 2 - 60720 p1 - 65856 p1 p10 - 93968 p1 p5 - 12885 p10 - 44960 p10 p5 2 / - 34540 p5 + 19660 p1 - 4925 p10 + 4850 p5 + 6500) / / 4 2 (-25 + 24 p1 + 20 p5 + 15 p10) + (576 p1 + 720 p1 p10 + 960 p1 p5 2 2 / + 225 p10 + 600 p10 p5 + 400 p5 - 576 p1 - 225 p10 - 400 p5) n / / 3 (-25 + 24 p1 + 20 p5 + 15 p10) and in Maple notation it is 5*(34560*p1^3+72576*p1^2*p10+91008*p1^2*p5+50220*p1*p10^2+126720*p1*p10*p5+ 79680*p1*p5^2+11475*p10^3+43650*p10^2*p5+55200*p10*p5^2+23200*p5^3-60720*p1^2-\ 65856*p1*p10-93968*p1*p5-12885*p10^2-44960*p10*p5-34540*p5^2+19660*p1-4925*p10+ 4850*p5+6500)/(-25+24*p1+20*p5+15*p10)^4+(576*p1^2+720*p1*p10+960*p1*p5+225*p10 ^2+600*p10*p5+400*p5^2-576*p1-225*p10-400*p5)/(-25+24*p1+20*p5+15*p10)^3*n The , 3, -th moment about the mean of of X(n), up to exponentially small error is 5 4 4 3 2 - 5 (19906560 p1 + 66686976 p1 p10 + 85598208 p1 p5 + 88957440 p1 p10 3 3 2 2 3 + 228925440 p1 p10 p5 + 147087360 p1 p5 + 59097600 p1 p10 2 2 2 2 2 3 + 228614400 p1 p10 p5 + 294451200 p1 p10 p5 + 126259200 p1 p5 4 3 2 2 + 19561500 p1 p10 + 101088000 p1 p10 p5 + 195696000 p1 p10 p5 3 4 5 4 + 168192000 p1 p10 p5 + 54144000 p1 p5 + 2581875 p10 + 16706250 p10 p5 3 2 2 3 4 5 + 43200000 p10 p5 + 55800000 p10 p5 + 36000000 p10 p5 + 9280000 p5 4 3 3 2 2 - 23639040 p1 - 32410368 p1 p10 - 60770304 p1 p5 + 2507760 p1 p10 2 2 2 3 - 37895040 p1 p10 p5 - 50664960 p1 p5 + 18030600 p1 p10 2 2 3 + 36968400 p1 p10 p5 + 7819200 p1 p10 p5 - 12556800 p1 p5 4 3 2 2 3 + 5983875 p10 + 23071500 p10 p5 + 30145500 p10 p5 + 14076000 p10 p5 4 3 2 2 + 952000 p5 - 42612480 p1 - 128622816 p1 p10 - 137766528 p1 p5 2 2 - 105593220 p1 p10 - 245898720 p1 p10 p5 - 138223680 p1 p5 3 2 2 3 - 23449500 p10 - 90686250 p10 p5 - 111111000 p10 p5 - 43612000 p5 2 2 + 76566960 p1 + 98924400 p1 p10 + 127477200 p1 p5 + 19411875 p10 2 + 66180000 p10 p5 + 47867500 p5 - 30222000 p1 - 4528125 p10 - 14487500 p5 / 6 4 3 ) / (-25 + 24 p1 + 20 p5 + 15 p10) - (331776 p1 + 829440 p1 p10 / 3 2 2 2 2 2 + 1105920 p1 p5 + 777600 p1 p10 + 2073600 p1 p10 p5 + 1382400 p1 p5 3 2 2 3 + 324000 p1 p10 + 1296000 p1 p10 p5 + 1728000 p1 p10 p5 + 768000 p1 p5 4 3 2 2 3 + 50625 p10 + 270000 p10 p5 + 540000 p10 p5 + 480000 p10 p5 4 3 2 2 2 + 160000 p5 - 304128 p1 - 336960 p1 p10 - 622080 p1 p5 - 64800 p1 p10 2 3 2 - 388800 p1 p10 p5 - 403200 p1 p5 + 16875 p10 - 90000 p10 p5 3 2 2 - 80000 p5 - 373248 p1 - 563760 p1 p10 - 670080 p1 p5 - 151875 p10 2 / - 435000 p10 p5 - 280000 p5 + 345600 p1 + 84375 p10 + 200000 p5) n / / 5 (-25 + 24 p1 + 20 p5 + 15 p10) and in Maple notation it is -5*(19906560*p1^5+66686976*p1^4*p10+85598208*p1^4*p5+88957440*p1^3*p10^2+ 228925440*p1^3*p10*p5+147087360*p1^3*p5^2+59097600*p1^2*p10^3+228614400*p1^2* p10^2*p5+294451200*p1^2*p10*p5^2+126259200*p1^2*p5^3+19561500*p1*p10^4+ 101088000*p1*p10^3*p5+195696000*p1*p10^2*p5^2+168192000*p1*p10*p5^3+54144000*p1 *p5^4+2581875*p10^5+16706250*p10^4*p5+43200000*p10^3*p5^2+55800000*p10^2*p5^3+ 36000000*p10*p5^4+9280000*p5^5-23639040*p1^4-32410368*p1^3*p10-60770304*p1^3*p5 +2507760*p1^2*p10^2-37895040*p1^2*p10*p5-50664960*p1^2*p5^2+18030600*p1*p10^3+ 36968400*p1*p10^2*p5+7819200*p1*p10*p5^2-12556800*p1*p5^3+5983875*p10^4+ 23071500*p10^3*p5+30145500*p10^2*p5^2+14076000*p10*p5^3+952000*p5^4-42612480*p1 ^3-128622816*p1^2*p10-137766528*p1^2*p5-105593220*p1*p10^2-245898720*p1*p10*p5-\ 138223680*p1*p5^2-23449500*p10^3-90686250*p10^2*p5-111111000*p10*p5^2-43612000* p5^3+76566960*p1^2+98924400*p1*p10+127477200*p1*p5+19411875*p10^2+66180000*p10* p5+47867500*p5^2-30222000*p1-4528125*p10-14487500*p5)/(-25+24*p1+20*p5+15*p10)^ 6-(331776*p1^4+829440*p1^3*p10+1105920*p1^3*p5+777600*p1^2*p10^2+2073600*p1^2* p10*p5+1382400*p1^2*p5^2+324000*p1*p10^3+1296000*p1*p10^2*p5+1728000*p1*p10*p5^ 2+768000*p1*p5^3+50625*p10^4+270000*p10^3*p5+540000*p10^2*p5^2+480000*p10*p5^3+ 160000*p5^4-304128*p1^3-336960*p1^2*p10-622080*p1^2*p5-64800*p1*p10^2-388800*p1 *p10*p5-403200*p1*p5^2+16875*p10^3-90000*p10*p5^2-80000*p5^3-373248*p1^2-563760 *p1*p10-670080*p1*p5-151875*p10^2-435000*p10*p5-280000*p5^2+345600*p1+84375*p10 +200000*p5)/(-25+24*p1+20*p5+15*p10)^5*n The limit of the scaled , 3, -th moment as n goes to infinity is 0 The , 4, -th moment about the mean of of X(n), up to exponentially small error is 7 6 6 (57330892800 p1 + 263722106880 p1 p10 + 342074327040 p1 p5 5 2 5 5 2 + 518665420800 p1 p10 + 1347275980800 p1 p10 p5 + 874296115200 p1 p5 4 3 4 2 + 565470720000 p1 p10 + 2205895680000 p1 p10 p5 4 2 4 3 + 2866544640000 p1 p10 p5 + 1240842240000 p1 p5 3 4 3 3 + 369165600000 p1 p10 + 1922227200000 p1 p10 p5 3 2 2 3 3 + 3751142400000 p1 p10 p5 + 3251404800000 p1 p10 p5 3 4 2 5 2 4 + 1056153600000 p1 p5 + 144342000000 p1 p10 + 940410000000 p1 p10 p5 2 3 2 2 2 3 + 2449440000000 p1 p10 p5 + 3188160000000 p1 p10 p5 2 4 2 5 6 + 2073600000000 p1 p10 p5 + 539136000000 p1 p5 + 31301437500 p1 p10 5 4 2 + 244944000000 p1 p10 p5 + 798255000000 p1 p10 p5 3 3 2 4 + 1386720000000 p1 p10 p5 + 1354320000000 p1 p10 p5 5 6 7 + 705024000000 p1 p10 p5 + 152832000000 p1 p5 + 2904609375 p10 6 5 2 4 3 + 26540156250 p10 p5 + 103882500000 p10 p5 + 225787500000 p10 p5 3 4 2 5 6 + 294300000000 p10 p5 + 230040000000 p10 p5 + 99840000000 p10 p5 7 6 5 + 18560000000 p5 + 206231961600 p1 + 1037808599040 p1 p10 5 4 2 4 + 1202727813120 p1 p5 + 2092204339200 p1 p10 + 4928432947200 p1 p10 p5 4 2 3 3 + 2883288268800 p1 p5 + 2183563008000 p1 p10 3 2 3 2 + 7814403072000 p1 p10 p5 + 9271535616000 p1 p10 p5 3 3 2 4 + 3645628416000 p1 p5 + 1252261620000 p1 p10 2 3 2 2 2 + 6037338240000 p1 p10 p5 + 10865767680000 p1 p10 p5 2 3 2 4 5 + 8649538560000 p1 p10 p5 + 2568760320000 p1 p5 + 375854175000 p1 p10 4 3 2 + 2284483500000 p1 p10 p5 + 5532840000000 p1 p10 p5 2 3 4 + 6672672000000 p1 p10 p5 + 4006176000000 p1 p10 p5 5 6 5 + 957670400000 p1 p5 + 46276312500 p10 + 339962062500 p10 p5 4 2 3 3 2 4 + 1037188125000 p10 p5 + 1681740000000 p10 p5 + 1528140000000 p10 p5 5 6 5 + 737648000000 p10 p5 + 147744000000 p5 - 1159374643200 p1 4 4 3 2 - 3946842132480 p1 p10 - 5014754979840 p1 p5 - 5149278345600 p1 p10 3 3 2 - 13368297369600 p1 p10 p5 - 8584104038400 p1 p5 2 3 2 2 - 3182908932000 p1 p10 - 12747087288000 p1 p10 p5 2 2 2 3 - 16769700864000 p1 p10 p5 - 7263953664000 p1 p5 4 3 - 914130967500 p1 p10 - 5078305260000 p1 p10 p5 2 2 3 - 10354523220000 p1 p10 p5 - 9221543040000 p1 p10 p5 4 5 4 - 3035708480000 p1 p5 - 93482353125 p10 - 693163406250 p10 p5 3 2 2 3 4 - 1980473625000 p10 p5 - 2752298250000 p10 p5 - 1871900000000 p10 p5 5 4 3 - 500543200000 p5 + 1538495873280 p1 + 3273826457088 p1 p10 3 2 2 2 + 4689082934784 p1 p5 + 2157893252640 p1 p10 + 6895546295040 p1 p10 p5 2 2 3 2 + 5168674815360 p1 p5 + 392916398400 p1 p10 + 2561189133600 p1 p10 p5 2 3 4 + 4533937564800 p1 p10 p5 + 2416044348800 p1 p5 - 19695396375 p10 3 2 2 3 + 107804286000 p10 p5 + 633226497000 p10 p5 + 900956264000 p10 p5 4 3 2 + 397278538000 p5 - 580846308480 p1 - 161687728800 p1 p10 2 2 - 798443918400 p1 p5 + 466248595500 p1 p10 + 502972788000 p1 p10 p5 2 3 2 - 144048508000 p1 p5 + 135116251875 p10 + 469025351250 p10 p5 2 3 2 + 424421135000 p10 p5 + 79003810000 p5 - 189537876000 p1 2 - 477201555000 p1 p10 - 472254740000 p1 p5 - 83396137500 p10 2 - 301008762500 p10 p5 - 195515425000 p5 + 125802912500 p1 / + 10380734375 p10 + 51575093750 p5 + 1897187500) / / 8 6 5 (-25 + 24 p1 + 20 p5 + 15 p10) + (191102976 p1 + 716636160 p1 p10 5 4 2 4 + 955514880 p1 p5 + 1119744000 p1 p10 + 2985984000 p1 p10 p5 4 2 3 3 3 2 + 1990656000 p1 p5 + 933120000 p1 p10 + 3732480000 p1 p10 p5 3 2 3 3 2 4 + 4976640000 p1 p10 p5 + 2211840000 p1 p5 + 437400000 p1 p10 2 3 2 2 2 + 2332800000 p1 p10 p5 + 4665600000 p1 p10 p5 2 3 2 4 5 + 4147200000 p1 p10 p5 + 1382400000 p1 p5 + 109350000 p1 p10 4 3 2 2 3 + 729000000 p1 p10 p5 + 1944000000 p1 p10 p5 + 2592000000 p1 p10 p5 4 5 6 + 1728000000 p1 p10 p5 + 460800000 p1 p5 + 11390625 p10 5 4 2 3 3 + 91125000 p10 p5 + 303750000 p10 p5 + 540000000 p10 p5 2 4 5 6 5 + 540000000 p10 p5 + 288000000 p10 p5 + 64000000 p5 + 852000768 p1 4 4 3 2 + 3110400000 p1 p10 + 3815424000 p1 p5 + 4447872000 p1 p10 3 3 2 2 3 + 11031552000 p1 p10 p5 + 6801408000 p1 p5 + 3129840000 p1 p10 2 2 2 2 2 3 + 11741760000 p1 p10 p5 + 14618880000 p1 p10 p5 + 6036480000 p1 p5 4 3 2 2 + 1087425000 p1 p10 + 5475600000 p1 p10 p5 + 10303200000 p1 p10 p5 3 4 5 + 8582400000 p1 p10 p5 + 2668800000 p1 p5 + 149596875 p10 4 3 2 2 3 + 946687500 p10 p5 + 2389500000 p10 p5 + 3006000000 p10 p5 4 5 4 3 + 1884000000 p10 p5 + 470400000 p5 - 3284748288 p1 - 7995386880 p1 p10 3 2 2 2 - 10765578240 p1 p5 - 6993540000 p1 p10 - 19289664000 p1 p10 p5 2 2 3 2 - 13122432000 p1 p5 - 2580093000 p1 p10 - 10997532000 p1 p10 p5 2 3 4 - 15361056000 p1 p10 p5 - 7046016000 p1 p5 - 331492500 p10 3 2 2 3 - 1966747500 p10 p5 - 4266945000 p10 p5 - 4032240000 p10 p5 4 3 2 2 - 1404880000 p5 + 2929734144 p1 + 3898510560 p1 p10 + 6280922880 p1 p5 2 2 + 1362436200 p1 p10 + 5119639200 p1 p10 p5 + 4348156800 p1 p5 3 2 2 3 + 99292500 p10 + 780187500 p10 p5 + 1607115000 p10 p5 + 968680000 p5 2 2 - 368409600 p1 + 345681000 p1 p10 - 107952000 p1 p5 + 146728125 p10 2 + 332737500 p10 p5 + 79800000 p5 - 319680000 p1 - 75515625 p10 / 7 2 - 178000000 p5) n / (-25 + 24 p1 + 20 p5 + 15 p10) + 3 (576 p1 / 2 2 + 720 p1 p10 + 960 p1 p5 + 225 p10 + 600 p10 p5 + 400 p5 - 576 p1 2 2 / 6 - 225 p10 - 400 p5) n / (-25 + 24 p1 + 20 p5 + 15 p10) / and in Maple notation it is (57330892800*p1^7+263722106880*p1^6*p10+342074327040*p1^6*p5+518665420800*p1^5* p10^2+1347275980800*p1^5*p10*p5+874296115200*p1^5*p5^2+565470720000*p1^4*p10^3+ 2205895680000*p1^4*p10^2*p5+2866544640000*p1^4*p10*p5^2+1240842240000*p1^4*p5^3 +369165600000*p1^3*p10^4+1922227200000*p1^3*p10^3*p5+3751142400000*p1^3*p10^2* p5^2+3251404800000*p1^3*p10*p5^3+1056153600000*p1^3*p5^4+144342000000*p1^2*p10^ 5+940410000000*p1^2*p10^4*p5+2449440000000*p1^2*p10^3*p5^2+3188160000000*p1^2* p10^2*p5^3+2073600000000*p1^2*p10*p5^4+539136000000*p1^2*p5^5+31301437500*p1* p10^6+244944000000*p1*p10^5*p5+798255000000*p1*p10^4*p5^2+1386720000000*p1*p10^ 3*p5^3+1354320000000*p1*p10^2*p5^4+705024000000*p1*p10*p5^5+152832000000*p1*p5^ 6+2904609375*p10^7+26540156250*p10^6*p5+103882500000*p10^5*p5^2+225787500000* p10^4*p5^3+294300000000*p10^3*p5^4+230040000000*p10^2*p5^5+99840000000*p10*p5^6 +18560000000*p5^7+206231961600*p1^6+1037808599040*p1^5*p10+1202727813120*p1^5* p5+2092204339200*p1^4*p10^2+4928432947200*p1^4*p10*p5+2883288268800*p1^4*p5^2+ 2183563008000*p1^3*p10^3+7814403072000*p1^3*p10^2*p5+9271535616000*p1^3*p10*p5^ 2+3645628416000*p1^3*p5^3+1252261620000*p1^2*p10^4+6037338240000*p1^2*p10^3*p5+ 10865767680000*p1^2*p10^2*p5^2+8649538560000*p1^2*p10*p5^3+2568760320000*p1^2* p5^4+375854175000*p1*p10^5+2284483500000*p1*p10^4*p5+5532840000000*p1*p10^3*p5^ 2+6672672000000*p1*p10^2*p5^3+4006176000000*p1*p10*p5^4+957670400000*p1*p5^5+ 46276312500*p10^6+339962062500*p10^5*p5+1037188125000*p10^4*p5^2+1681740000000* p10^3*p5^3+1528140000000*p10^2*p5^4+737648000000*p10*p5^5+147744000000*p5^6-\ 1159374643200*p1^5-3946842132480*p1^4*p10-5014754979840*p1^4*p5-5149278345600* p1^3*p10^2-13368297369600*p1^3*p10*p5-8584104038400*p1^3*p5^2-3182908932000*p1^ 2*p10^3-12747087288000*p1^2*p10^2*p5-16769700864000*p1^2*p10*p5^2-7263953664000 *p1^2*p5^3-914130967500*p1*p10^4-5078305260000*p1*p10^3*p5-10354523220000*p1* p10^2*p5^2-9221543040000*p1*p10*p5^3-3035708480000*p1*p5^4-93482353125*p10^5-\ 693163406250*p10^4*p5-1980473625000*p10^3*p5^2-2752298250000*p10^2*p5^3-\ 1871900000000*p10*p5^4-500543200000*p5^5+1538495873280*p1^4+3273826457088*p1^3* p10+4689082934784*p1^3*p5+2157893252640*p1^2*p10^2+6895546295040*p1^2*p10*p5+ 5168674815360*p1^2*p5^2+392916398400*p1*p10^3+2561189133600*p1*p10^2*p5+ 4533937564800*p1*p10*p5^2+2416044348800*p1*p5^3-19695396375*p10^4+107804286000* p10^3*p5+633226497000*p10^2*p5^2+900956264000*p10*p5^3+397278538000*p5^4-\ 580846308480*p1^3-161687728800*p1^2*p10-798443918400*p1^2*p5+466248595500*p1* p10^2+502972788000*p1*p10*p5-144048508000*p1*p5^2+135116251875*p10^3+ 469025351250*p10^2*p5+424421135000*p10*p5^2+79003810000*p5^3-189537876000*p1^2-\ 477201555000*p1*p10-472254740000*p1*p5-83396137500*p10^2-301008762500*p10*p5-\ 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