I am struggling to find the expectation of a derivate of the Coupon collector's problem
In the description of my problem, I will write down some number but each of them are not fix, I am searching a general solution
There is 4 packets:
- A with 40 coupons
- B with 30 coupons
- C with 20 coupons
- D with 10 coupons
All the 100 coupons are differents.
At each draw, we draw a coupon from a random packet. And then put the coupon back in this packet.
In the random choice of the packet, each packet have it own probability to be selected:
- A is selected 60% of the time
- B is selected 20% of the time
- C is selected 15% of the time
- D is selected 5% of the time
My problem is to calculate the probability to see all the 100 coupons in n draws. And the expectation of the number of draw needed to se all the 100 coupons
It seem related to this question: Expected number of rolls for an unfair die to get all possibile values at least once (it's the problem with 2 packet, one with 2,3,4,5,6 and one with 1, the first packet have 5/7 to be picked and the second 2/7) But the solution doesn't have any explanation and I have difficulty to understand from where it come
Thank you
The part 3.2 of this document bring an answer and a formula to calculate the expectation
https://mat.uab.cat/matmat_antiga/PDFv2014/v2014n02.pdf