I mimicked 1 millions completions of OSRS directors and graphed the completion kc's. This way it is possible to see the spread of completions. Since completion can be somewhat vague I mimicked a couple of distinct choices. Default getting 1 of drops unique into the boss assumed excluding any jars/pets. Calculating average conclusion is hard to do when 2 drops have different weighting. It also does not show the variance, which I suppose you need to have the ability to calculate somehow but I am not certain how.
In this instance it simplifies because the time to collect a RuneScape gold with already received m-1 of then follows a geometric distribution. But even then types don't have closed. Therefore the general case (namely this one) is sort of hopeless.
On is take all arranged lists of numbers 1,.,n of a certain length (this is kc) with repetitions where every number happens at least once together with the previous number occurring exactly once, assigned weights, and summed. Assuming you've got an easy enough tactics to generate the lists, this will be an computation. (An easy means to do so is to take a list of length l-1 with repeats in the numbers without I at which every other number seems at least once and then append I to the conclusion ).
Additionally, it is worth noting the complicated parts of the coupon collector's problem only arise because the conventional difficulty attempts to calculate the mean, variance, as well as the pmf right out of a pmf using analytic methods, which gets real ugly since there are many nested, iterative calculations. Computational calculation going pmf of single item – cdf of only item – cdf of – pmf of is a lot of dull impossible to do by hand math, but is computationally very easy. This can be quickly calculated by A computer. See my comment above for the exact solution that is computational. It will get rs3 gold to match the image above, although this computational solution won't get you the variance or mean.
That said this computational method has incredibly low error except in cases under 1/100 of kc. I mean the points on your curves. Everything you did (I assume) is you sampled a known distribution. Alternatively, you could have plotted the same distribution! Ah right, it might have made sense to get it done like that. Though I think combining the distrubutions would have contributed slightly skewed results on more prevalent drops as you can't get 2 distinct drops at the exact same time. I would not understand how to account for this.
