More coupon collecting - some playing about
I've been playing about with the asymptotic distribution a bit.
From last time,
P(W <= w) ~= exp[-n . exp(-w/n)]
Rather than work with W, the total number of coupons collected, in some ways its easier to work with R = W/n, what I will call the collection ratio -- the number collected per distinct coupon.
i.e.
P(R <= r) ~= exp[-n . exp(-r)]
= exp[- exp(-{r-log(n)})]
That's a Gumbel (Fisher-Tippett) distribution. Well, it already was when it was in terms of W, but it's a bit easier to see now. That the process of collecting until you have all possible tokens might yield an extreme value distribution is not especially suprising.
We immediately have for this asymptotic distribution for R that the location parameter is log(n) and the scale parameter is 1.
Consequently we have (asymptotically) for R,
mean: log(n) + γ
median: log(n) - log(log(2)) (NB -log(log(2)) is positive)
mode: log(n)
variance: π2/6 (now that's familiar!)
I have some more to add but I have to leave it there for now.
From last time,
P(W <= w) ~= exp[-n . exp(-w/n)]
Rather than work with W, the total number of coupons collected, in some ways its easier to work with R = W/n, what I will call the collection ratio -- the number collected per distinct coupon.
i.e.
P(R <= r) ~= exp[-n . exp(-r)]
= exp[- exp(-{r-log(n)})]
That's a Gumbel (Fisher-Tippett) distribution. Well, it already was when it was in terms of W, but it's a bit easier to see now. That the process of collecting until you have all possible tokens might yield an extreme value distribution is not especially suprising.
We immediately have for this asymptotic distribution for R that the location parameter is log(n) and the scale parameter is 1.
Consequently we have (asymptotically) for R,
mean: log(n) + γ
median: log(n) - log(log(2)) (NB -log(log(2)) is positive)
mode: log(n)
variance: π2/6 (now that's familiar!)
I have some more to add but I have to leave it there for now.