# Thread: Simple but quite confusing Problem

1. ## Simple but quite confusing Problem

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).

2. ## Re: Simple but quite confusing Problem

In part b) you select with replacement, and therefore those 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 . For the variance part, now you need to calculate the covariance terms .

The tricky part here is to calculate the , and the key is to recognize that as , the selections 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:

Spoiler:

Simplify it and you should obtain the answer.

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