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Draw independent Uniform(0,1) numbers until their running sum exceeds 1. What is the expected number of draws?

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Starting from 0, repeatedly add a random number sampled uniformly from [0,1]. What is the expected number of samples needed for the running sum to exceed 2?Independent uniform(0,1) draws are shown one at a time. What is the expected index of the first draw that is not the smallest so far?n independent uniform draws are shown one at a time and you may accept only one, at the moment it is shown. What stopping rule maximises the expected value of the draw you accept?

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