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If we let $U_1, U_2, U_3,..., U_n$ be uniform (0,1), find $$\mathbb E[\sum_{i=0}^n iU_i^{i-1}]$$which, using the linearity of expectation, gives $$\sum_{i=0}^n \mathbb E[i U_i^{i-1}]$$

Doing this summation gives us $$\Bbb E[U_1^0]+\Bbb E[2U_2^1]+\Bbb E[3U_3^2]+...+\Bbb E[nU_n^{n-1}]$$$$=1+1+1+...$$ Which is equal to $\infty$, or rather equals infinty in the limit $n\to \infty$.

How can an expected value be infinte? How would one give this answer if asked to 'find' the expected value? Does this mean that the expected value does not exist? Or have I miscalculated this - it seems unlikely that a finite sum should give an infinite answer.

mathenthusiast
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  • How did you obtain an infinite value from a finite sum of ones?? The sudden appearance of the ellipsis at the end of your final expression has no valid justification. – whuber Apr 26 '18 at 12:36
  • If $n$ is arbitrary is it not possible that it can tend toward infinity? – mathenthusiast Apr 26 '18 at 12:37
  • An "arbitrary" or unknown value does not imply a limit or any kind of "tendency to infinity." Perhaps a limit was present in your original problem, but so far it hasn't appeared in your question. If it is intended, then the issue has nothing to do with expectation, because you state that all expectations equal $1$, which is finite. – whuber Apr 26 '18 at 12:38
  • There was no limit so obviously I'm making a leap that I shouldn't; how would you calculate this sum? EDIT: Is the answer simply $n$? – mathenthusiast Apr 26 '18 at 12:38

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