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Expectation as a sum of tails

MediumRandom VariablesExpected Value~4m

For a non-negative integer-valued XX, which identity expresses E[X]E[X] using only tail probabilities?

Show hints (2)+
  1. Write k as a sum of 1's and swap summation order.
  2. Sum the survival probabilities P(X > k).

Answer

Reveal answer →

E[X]=∑k≥0P(X>k)E[X] = \sum_{k\ge 0} P(X>k)

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