How many boxes of cereal do you expect to have to buy to get all N toys?

Suppose ๐ต different toys are offered in boxes of a favorite brand of cereal. You want to collect all the different toys. How many boxes of cereal do you expect to have to buy to get all ๐ต toys? Let the random variable ๐ฟ be the number of boxes of cereal you need to buy to get each toy at least once. Also, assume it is equally likely that any one of the ๐ต toys will be in each box of cereal.

a. Show that $$๐ฟ = ๐ฟ_๐ + ๐ฟ_๐ + ๐ฟ_๐ + โฏ + ๐ฟ_{๐ตโ๐}$$ , where $$๐ฟ_๐$$ has a geometric distribution with probability $$\frac{๐ตโ๐}{๐ต}$$.

Hint: consider the random variable which is the number of boxes needed to get the ๐๐๐ toy after getting ๐ โ ๐ of all of them.

b. Find ๐ฌ(๐ฟ). The correct answer indicates that to get all eight toys offered with the current MacDonalds Happy Meal, you would, under the assumptions of the calculation, expect to buy about $$22$$ Happy Meals.

I was able to calculate part b as $$E(X) = N(\frac{1}{N} + \frac{1}{N-1} + ... + \frac{1}{N - (N-1)})$$ but I can't figure out how to start part a.

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• There is literature on this. It's called the Coupon Collector's Problem. – Gerry Myerson Feb 12 at 2:58
• I found information on the coupon collector's problem before, but I haven't been able to properly link it to what part a of the question is asking me to do. I was able to use it to solve part b, but whatever material I read on the problem doesn't clear up part a at all for me. – hmtkd Feb 12 at 3:16