Question

. A box contains 2n
tickets among which nei tickets bear the number ( = 0, 1,
2, … , ). A group of m tickets is drawn. Let S denote the sum of their numbers.
Find E(S) and Var S.

Answer 1

Answer:

There are 2n tickets in a jar.The frequency of the tickets of number i is (ni) where i=0,1,2,..n.

m tickets are drawn randomly without replacement.

Let S be the sum of the numbers drawn.Find E(S) and variance of S.This problem is a type of coupon collector without replacement where there are (nj) tickets of type j and we ask about the expectation of the sum of the ticket values after m tickets have been drawn. Using the methodology from the following two MSE links we find that the EGF by multiplicity of a set of coupons of type j is given by

∑k=0(nj)(nj)k––zkk!=(1+z)(nj).

Distributing all n types of coupons we get

∏j=0n(1+z)(nj)=(1+z)2n

for a total count according to multiplicity of

m2n=m!×(2nm).

Marking the contribution of a ticket of type j with uj we obtain the mixed generating function

G(z,u)=∏j=0n(1+ujz)(nj).

Differentiate and evaluate at u=1 to obtain

∂∂uG(z,u)∣∣∣u=1=∏j=0n(1+ujz)(nj)∑j=0n(1+ujz)−(nj)(nj)(1+ujz)(nj)−1juj−1z∣∣∣∣u=1=(1+z)2n∑j=1n(nj)jz1+z=z(1+z)2n−1∑j=1nj(nj)=z(1+z)2n−1∑j=1nn(n−1j−1)=n2n−1z(1+z)2n−1.