- #1
suyver
- 248
- 0
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:
[tex]\left< \sum_{i=1}^\infty p_i \right>=1[/tex]
Does anybody know a proof as to why this is the case?
[tex]\left< \sum_{i=1}^\infty p_i \right>=1[/tex]
Does anybody know a proof as to why this is the case?