In the coupon collector's problem, instead of setting E(X) as the amount of time required to select a new distinct Coupon, if we set it as the expectant amount of required to select a given coupon type, then using linearity of expectation, this can be written as
E(Coupon_type_1 + Coupon_type_2 + Coupon_type_3 ....) = E(Coupon_type_1) + E(Coupon_type_2) +... E(Coupon_type_n).
As the Expectancy of each of eventually getting each of these coupons is then 1/p = n, the result of equation would lead us to E(X) = (n + n + ...) = n * n, (which is not ≈ n*log(n) ).
I know this isn' t the correct approach, can anyone help in explaining why this approach is wrong while using linearity of expectation ?
If $C$ denotes the time needed to collect all coupons, and $C_i$ the time needed to collect coupon $C_i$ then:$$C=\max(C_1,\dots,C_n)$$so not: $$C=C_1+\cdots+C_n$$