Expected number of types of coupons collected after $n$ picks

Question

There are $k$ types of coupons. Independently of the types of previously collected coupons, each new coupon collected is of type $i$ with probability $p_i$ s.t. $\sum\limits_ {i=1}^kp_i =1$.

If $n$ coupons are collected, find the expected number of distinct types that appear in this set. (That is, find the expected number of types of coupons that appear at least once in the set of $n$ coupons.)

Answer 1

The types of the successively collected coupons are independent hence each event $A_i$ that no coupon of type $i$ was collected has probability $$\mathrm P(A_i)=(1-p_i)^n$$ The number $N$ of types of coupons collected is the number of events $A_i$ not realized hence

$$ \mathrm E(N)=\sum\limits_{i=1}^k(1-\mathrm P(A_i))=k-\sum\limits_{i=1}^k(1-p_i)^n $$

Sanity checks:

• For $n=1$, $\mathrm E(N)=1$ (explain why).
• For $n=2$, $\mathrm E(N)=2-\sum\limits_{i=1}^kp_i^2$ (describe the event of probability $\sum\limits_{i=1}^kp_i^2$ involved).
• For every $k\geqslant2$ the function $n\mapsto\mathrm E(N)$ is increasing (explain why).
• And finally, when $n\to\infty$, $\mathrm E(N)\to k$ (explain why).