Suppose 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 toys? Let the random variable 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 toys will be in each box of cereal.
a. Show that $ = _ + _ + _ + ⋯ + _{−}$ , where $_$ has a geometric distribution with probability $\frac{−}{}$.
Hint: consider the random variable which is the number of boxes needed to get the toy after getting − of all of them.
b. Find (). 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.