# Simple but quite confusing Problem

#### sk6398

##### New Member
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
$$S = \sum_{i=1}^n X_i$$
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:
$$\frac {1} {N(N-1)} \left[\left(\sum_{i=1}^n i\right)^2 - \sum_{i=1}^n i^2 \right]$$