Coupon collector's problem

Coupon collector's problem
Parameters	${\displaystyle n\in \mathbb {N} }$ – number of faces on die
Support	${\displaystyle k\in \mathbb {N} }$ – rolls taken for all faces to appear
PMF	${\displaystyle {\frac {(n-1)^{\{k-1\}}}{n^{k-1}}}}$
CDF	${\displaystyle {\frac {n^{\{k\}}}{n^{k}}}}$
Mean	${\displaystyle nH_{n}}$
Variance	${\displaystyle n^{2}H_{n}^{(2)}-nH_{n}}$
Skewness	${\displaystyle {\frac {2n^{3}H_{n}^{(3)}-3n^{2}H_{n}^{(2)}+nH_{n}}{\left(n^{2}H_{n}^{(2)}-nH_{n}\right)^{3/2}}}\ {\underset {n}{\sim }}\ 6^{3/2}2{\frac {\zeta (3)}{\pi ^{3}}}}$
Excess kurtosis	${\displaystyle {\frac {6n^{4}H_{n}^{(4)}-12n^{3}H_{n}^{(3)}+7n^{2}H_{n}^{(2)}-nH_{n}}{(n^{2}H_{n}^{(2)}-nH_{n})^{2}}}\sim {\frac {6}{5}}}$
MGF	${\displaystyle {{\frac {n}{e^{t}}} \choose n}^{-1}}$
CF	${\displaystyle {{\frac {n}{e^{it}}} \choose n}^{-1}}$
PGF	${\displaystyle G(z)={{\frac {n}{z}} \choose n}^{-1}}$

In probability theory, the coupon collector's problem refers to mathematical analysis of "collect all coupons and win" contests. It asks the following question: if each box of a given product (e.g., breakfast cereals) contains a coupon, and there are n different types of coupons, what is the probability that more than t boxes need to be bought to collect all n coupons? An alternative statement is: given n coupons, how many coupons do you expect you need to draw with replacement before having drawn each coupon at least once? The mathematical analysis of the problem reveals that the expected number of trials needed grows as ${\displaystyle \Theta (n\log(n))}$. For example, when n = 50 it takes about 225 trials on average to collect all 50 coupons. Sometimes the problem is instead expressed in terms of an n-sided die.