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    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).

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    Re: Simple but quite confusing Problem


    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[S]. 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:

    Spoiler:


    Simplify it and you should obtain the answer.

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