I choose a random number [itex]p_1 \in [0,1)[/itex] and a subsequent series of (increasingly smaller) random numbers [itex]p_i \in [0, p_{i-1})[/itex]. Then I can calculate the sum [itex]\sum_{i=1}^\infty p_i[/itex]. Naturally, this sum is dependent on the random numbers chosen, so its particular result is not very insightful. However, it appears that its mean is rather surprising:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\left< \sum_{i=1}^\infty p_i \right>=1[/tex]

Does anybody know a proof as to why this is the case?

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# Mean of sum of random numbers

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