It's been awhile since I worked with problems related to the coupon-collector problem. I believe the distribution of collection times is a combination of various geometric distributions and itself is not from a common distribution family. I think the distribution values (as a formula) may be available.
Since this is fairly easy to do, and in the absence of such distribution formulas, I ran a simulation of 1 million trials of the equal-probability coupon collector case with 10 items. I chose 10 so that the simulation would not take very long. Maybe I'll do the same thing for 169 but the time it will take will be much longer and presenting all the output will be more arduous.
Note EV(number of selections to select all 10) = 10*H(10) = 29.28968
Number of Selections Required | Tally | Cumulative Tally |
---|
1 | 0 | 0 |
2 | 0 | 0 |
3 | 0 | 0 |
4 | 0 | 0 |
5 | 0 | 0 |
6 | 0 | 0 |
7 | 0 | 0 |
8 | 0 | 0 |
9 | 0 | 0 |
10 | 328 | 328 |
11 | 1699 | 2027 |
12 | 4267 | 6294 |
13 | 8146 | 14440 |
14 | 13003 | 27443 |
15 | 18629 | 46072 |
16 | 24322 | 70394 |
17 | 29475 | 99869 |
18 | 34734 | 134603 |
19 | 38587 | 173190 |
20 | 41420 | 214610 |
21 | 43791 | 258401 |
22 | 44498 | 302889 |
23 | 45066 | 347965 |
24 | 44679 | 392644 |
25 | 43444 | 436088 |
26 | 42569 | 478657 |
27 | 40356 | 519013 |
28 | 39047 | 558060 |
29 | 36782 | 594842 |
30 | 34464 | 629306 |
31 | 32324 | 661630 |
32 | 29856 | 691486 |
33 | 27476 | 718962 |
34 | 25584 | 744546 |
35 | 23211 | 767757 |
36 | 21372 | 789129 |
37 | 19857 | 808986 |
38 | 17921 | 826907 |
39 | 16266 | 843173 |
40 | 14865 | 858038 |
41 | 13732 | 871770 |
42 | 12396 | 884166 |
43 | 11031 | 895197 |
44 | 10294 | 905491 |
45 | 9123 | 914614 |
46 | 8346 | 922960 |
47 | 7525 | 930485 |
48 | 6900 | 937385 |
49 | 6017 | 943402 |
50 | 5569 | 948971 |
51 | 5082 | 954053 |
52 | 4563 | 958616 |
53 | 4073 | 962689 |
54 | 3743 | 966432 |
55 | 3369 | 969801 |
56 | 3011 | 972812 |
57 | 2732 | 975544 |
58 | 2417 | 977961 |
59 | 2202 | 980163 |
60 | 2039 | 982202 |
61 | 1798 | 984000 |
62 | 1569 | 985569 |
63 | 1389 | 986958 |
64 | 1220 | 988178 |
65 | 1209 | 989387 |
66 | 1110 | 990497 |
67 | 959 | 991456 |
68 | 825 | 992281 |
69 | 801 | 993082 |
70 | 696 | 993778 |
71 | 601 | 994379 |
72 | 558 | 994937 |
73 | 492 | 995429 |
74 | 451 | 995880 |
75 | 370 | 996250 |
76 | 378 | 996628 |
77 | 352 | 996980 |
78 | 288 | 997268 |
79 | 248 | 997516 |
80 | 260 | 997776 |
81 | 225 | 998001 |
82 | 198 | 998199 |
83 | 204 | 998403 |
84 | 151 | 998554 |
85 | 120 | 998674 |
86 | 130 | 998804 |
87 | 136 | 998940 |
88 | 118 | 999058 |
89 | 93 | 999151 |
90 | 82 | 999233 |
91 | 67 | 999300 |
92 | 67 | 999367 |
93 | 68 | 999435 |
94 | 62 | 999497 |
95 | 51 | 999548 |
96 | 52 | 999600 |
97 | 31 | 999631 |
98 | 34 | 999665 |
99 | 38 | 999703 |
100 | 29 | 999732 |
101 | 23 | 999755 |
102 | 25 | 999780 |
103 | 15 | 999795 |
104 | 24 | 999819 |
105 | 17 | 999836 |
106 | 21 | 999857 |
107 | 14 | 999871 |
108 | 18 | 999889 |
109 | 13 | 999902 |
110 | 10 | 999912 |
111 | 9 | 999921 |
112 | 5 | 999926 |
113 | 4 | 999930 |
114 | 5 | 999935 |
115 | 8 | 999943 |
116 | 6 | 999949 |
117 | 3 | 999952 |
118 | 9 | 999961 |
119 | 4 | 999965 |
120 | 4 | 999969 |
121 | 4 | 999973 |
122 | 1 | 999974 |
123 | 2 | 999976 |
124 | 0 | 999976 |
125 | 3 | 999979 |
126 | 5 | 999984 |
127 | 3 | 999987 |
128 | 1 | 999988 |
129 | 1 | 999989 |
130 | 2 | 999991 |
131 | 1 | 999992 |
132 | 0 | 999992 |
133 | 1 | 999993 |
134 | 1 | 999994 |
135 | 0 | 999994 |
136 | 0 | 999994 |
137 | 1 | 999995 |
138 | 1 | 999996 |
139 | 1 | 999997 |
140 | 0 | 999997 |
141 | 0 | 999997 |
142 | 1 | 999998 |
143 | 0 | 999998 |
144 | 1 | 999999 |
145 | 0 | 999999 |
146 | 0 | 999999 |
147 | 0 | 999999 |
148 | 0 | 999999 |
149 | 0 | 999999 |
150 | 0 | 999999 |
151 | 0 | 999999 |
152 | 0 | 999999 |
153 | 0 | 999999 |
154 | 0 | 999999 |
155 | 1 | 1000000 |
Obviously, any pertinent cumulative percentiles can be easily read off the table.
Last edited by whosnext; 09-06-2018 at 07:30 PM.