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Nice problem. I'd like to generalize it as follows:Let X(1), X(2), ... be independent and identically distributed uniform random variables over [0,1]. Let y be any real number. Set N(y)=min{n: X(1)+X(2)+...+X(n)>y}. Find expected value E[N(y)]. As has been shown, E[N(1)]=e.If we let f(y)= E[N(y)], then it can be shown that f(y)= 1 + INTEGRAL{f(u) [where u runs from max(0,y-1) to y]}.The challenge now is to solve this integral equation. Any takers?
Nice problem. I'd like to generalize it as follows:
Let X(1), X(2), ... be independent and identically distributed uniform random variables over [0,1]. Let y be any real number. Set N(y)=min{n: X(1)+X(2)+...+X(n)>y}. Find expected value E[N(y)]. As has been shown, E[N(1)]=e.
If we let f(y)= E[N(y)], then it can be shown that f(y)= 1 + INTEGRAL{f(u) [where u runs from max(0,y-1) to y]}.
The challenge now is to solve this integral equation. Any takers?
