The coupon collector's problem asks for the number of independent and identically distributed draws needed to observe every one of 
types of object, where 
is a positive integer.
On each draw, type 
has probability 
, with
 |
The collection time is the first draw at which all types have appeared. If any 
,
this time is infinite. For the discrete
uniform distribution, 
, and the expectation
value of the collection time is 
, where 
is a harmonic number.
For repeated complete collections, let 
be the number of independent uniform draws required
until each of the 
types has appeared at least 
times. Castro and Marinho (2026) determined a Gumbel
distribution limit as 
. Let 
be the distribution
function of the gamma distribution with
shape 
and scale 1, and set 
. If 
tends to 0, a finite positive value 
, or infinity, respectively, the window 
is 
, 
with 
, or 
with 
. Here 
is the negative real branch of the Lambert
W-function. For every real 
,
 |
and 
,
where 
is the Euler-Mascheroni constant (Castro
and Marinho 2026).
