Question

There are n types of coupons. Each newly obtained coupon is, independently, type i with probability p i , i = 1,...,n. Find the expected number and the variance of the number of distinct types obtained in a collection of k coupons

Answer 1

Answer:

There are n types of coupons. Each newly obtained coupon is, independently, type i with probability p i , i = 1,...,n. Find the expected number and the variance of the number of distinct types obtained in a collection of k coupons

Explanation: The solution for the expectation has already been given in the comments. The calculation is slightly simplified by considering the number. Y= n-X of coupon types not collected with E[X] =N-E[Y] and Var(X) = Var(Y). Let Yi denote the indicator variable

Type i is not obtained, its expectations is the probability (1-pi)^k

Of not obtaining a coupon of type i, so the expected number of coupons is not obtained is

E[Y]= E{£iYi}= Ei(1-pi)^k

The variance is calculated analogously by expressing in terms of expectation,

Var(Y)=E{[EiYi]}^2-E[EiYi]^2

= Ei(1-pi)^k(1-(1-pi)^k + Ei=/j(1-pi-pj)^k-(1-pi)^k(1-pj)^k