Can't understand why all the expectations are added from 1 to n?

#1

In Coupon Collector Problem the random variable X(i) is defined as the expectation of i distinct coupons. So for calculating the number of expected packets containing n distinct coupon should not it be X(n) instead of summing all the expectations from 1 to n. I don’t understand why all the expectations have been added.

#2

Xi is defined as the number of lots bought before collecting ith coupon…rather than i coupons

#3

Ok Xi is the number of items bought before collecting ith coupon. Then why expectations have been added. Shouldn’t it be expectation of Xn because Xn is the number of items bought before collecting nth distinct coupons. Please explain.

#4

The coupon collector problem asks for the number of lots to be bought before we have a coupon of each type…so X1 is number of lots bought for 1st new coupon…X2 is lots bought before obtaining 2nd coupons…and so on…so, to get all coupons…we need to sum x1 + x2 + …xn.

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#5

Thanks sir for explanation!!