Expected number of rejected null hypotheses using FDR

Problem:

Let X1,X2,…,X500X_1, X_2, \dots, X_{500} be independently identically distributed. For a constant aa, suppose we know the probability P(Xi≤k∗a) ∀k=1,…,500.P(X_i\leq k*a)\ \forall k = 1,\dots , 500.

We now sort the XX’s so that X(1)≤X(2)≤⋯≤X(500)X_{(1)}\leq X_{(2)}\leq \dots\leq X_{(500)}.

Find the expectation E(K)E(K), K=max(k)K = \max(k) where kk satisfies X_{(k)}\leq k*a.X_{(k)}\leq k*a.

My attempt:

I don’t really have any rigorous proof, but I simply think of this as an “average” problem, and so the answer would be E(K) = \sum_{k=1}^{500}P(X_i\leq k*a).E(K) = \sum_{k=1}^{500}P(X_i\leq k*a).

Note:
You may notice that this problem relates to false discovery rate. I have been banging my head against the wall for couple days and can’t seem to get anywhere. Any help/suggestions/ideas are much appreciated!

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