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Finiteness of a converging random number series 
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#1
Feb2912, 12:54 AM

P: 3,408

1. Imagine a positive point x not equal to zero.
2. Consider a randomly chosen point y with distance to zero less than x. 3. Let y=x. Repeat #2. 4. Is the sum of the yvalues finite as y approaches zero? 


#2
Feb2912, 02:11 AM

P: 15

I don't know if it always converges, but on average it converges to x (by "average" I mean that for any given random y value, the average of all choices is x/2, so y, on average, equals x/2). 


#3
Feb2912, 05:41 AM

P: 144

Hi, I understand this as follows: denote by ran(x) a random number between 0 and x. Let x_{1}=ran(1), and let x_{i}=ran(x_{i1}) for x>1.
Let S be the sum Ʃx_{i}. As noted above, the expected value of S is 1 (does require a very minor argument). The chance of the series not converging is 0. For example the chance of S>N must be less than 1/N, for the average sum to be 1, so the chance of divergence is less than 1/N for any positive N. (A small simulation shows that the chance of the sum exceeding 7 is about 1 in 10 million) 


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