Question

Question 3. There are 100 types of coupons. Independently of the types of previously collected coupons, each new coupon is equally likely to be of any of the 100 types. If 200 coupons are collected, let X be the number of types of coupons that appear exactly twice in this collection. Find E[X] and Var(X) by using indicator variables and linearity of expectation.

Answer 1

Let, each new coupon collected of type i has probability p_i .  

Define, Random Variable:

where 1 <= i <= 100

Probability that exactly 2 coupons of type i are collected from 200 coupons:

Let

where X is random variable denotes the number of types of coupons that appear exactly twice in collection of 200 coupons.

So,

Expectaton:

Variance:

Note: X_i's are independent random variables.