Show that if X is a continuous r.v. and it takes only positive values then $E(X) =\int_{0}^{∞}Z P[X ≥ t] dt $

Asked May 09 '16 at 21:49

Active May 11 '16 at 17:44

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Show that if X is a continuous r.v. and it takes only positive values then: $$E(X) =\int_{0}^{∞} P[X ≥ t] dt$$

I am not really sure how to begin this proof. Any help or insight would be appreciated.

edited May 11 '16 at 17:44

Henry

asked May 09 '16 at 21:49

Tejava

Consider the case where $X$ takes on only finitely many values for intuition. You'll find that you want to interchange the summation given by the integration and the summation given by the probability. Now justify this interchange in general. This is one approach. Another way is to work directly with the definitions in terms of partitions. – Ian May 09 '16 at 21:52

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