Let $U_1,...,U_n$ are i.i.d uniformly distributed on $[0,1]$. Define $$N=\min\{n\geq 0:\sum_{i=1}^{n}U_i>\frac{1}{2}\}$$
Compute $EN$
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You know something about martingales and stopping times? – Dominik Kutek Oct 05 '20 at 17:49
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@DominikKutek Yes, I know, but this doesn't help here. – Darek Oct 05 '20 at 17:59
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1Does this help? https://math.stackexchange.com/q/111314/601852 – Dominik Kutek Oct 05 '20 at 18:05
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you're right, martingale theory only gives us bound $\mathbb E[N] \in [1,3]$ and can help us find $\mathbb E[ \sum_{i=1}^N U_i]$ if we already have $\mathbb E[N]$. Sorry, I was wrong, but as it seems in the shared question, you can find distribution of $N$ by just integrating scaled symplex – Dominik Kutek Oct 05 '20 at 18:08
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Thank you @DominikKutek. This helps me. – Darek Oct 05 '20 at 18:34