Simple but quite confusing Problem

#1
Hi all, I'm solving the problems in probability but, the question below looks simple but quite confusing to me.

Regarding question 2-(c), any tips or advice would be appreciated to me.


2. Suppose that we have N balls numbered 1 to N. If we let Xi be the number on the ith drawn
ball so that P(Xi = k) = 1/N for k = 1, • • • ,N.

(a) E(Xi), V (Xi) =?

(b) Let S be the sum of the numbers on n balls selected at random, with replacement, from
1 to N. Calculate E(S) and V (S).

(c) Let S be the sum of the numbers on n balls selected at random, without replacement,
from 1 to N. Calculate E(S) and V (S).
 

BGM

TS Contributor
#2
Assume you already know about part b) and have the expression

\( S = \sum_{i=1}^n X_i \)

In part b) you select with replacement, and therefore those \( X_i \) are independent, and you should calculate the answer with ease.

However, in part c) they are dependent; but they still have the identical distribution. So it should not affect your calculation of \( E \). For the variance part, now you need to calculate the covariance terms \( Cov[X_i, X_j] \).

The tricky part here is to calculate the \( E[X_iX_j] \), and the key is to recognize that as \( i \neq j \), the selections \( X_i, X_j \) cannot be the same number. When you try to do the multiplication, it is like a square matrix missing the diagonal. And you should obtain something similar to the following:

\( \frac {1} {N(N-1)} \left[\left(\sum_{i=1}^n i\right)^2 - \sum_{i=1}^n i^2 \right] \)

Simplify it and you should obtain the answer.