Let $X_1, X_2, \dots$ be IID Unif$(0,1)$ random variables and let $N=\min \{n : S_n=X_1 + \dots + X_n > \ln(2) \}$. Find the expectation of $N$.
I've tried three approaches.
- First I showed that $S_n- \frac n2$ is a martingale and wanted to give Optional Stopping Theorem (if it holds) but it would give $$\Bbb E(S_N - \frac N2)=\Bbb E(S_1 - \frac 12)=0 \iff 2\Bbb E(S_N) = \Bbb E( N) $$
Then I know that $S_N \in (\ln(2) , \ln(2)+1)$ almost surely but nothing useful from it.
My second approach was similar. I was considering random variables Unif$(-1,1)$ (their absolute value is Unif$(0,1)$) and looking at $S_n^2- \frac n3$ which is a martingale and use Optional Stopping Theorem (if it holds). But this gives useless results too.
My third approach was just to compute $\Bbb P(N=k)$ and try to identify a pattern to show by induction. I have that $\Bbb P(N=1)= 1-\ln(2)$ but $\Bbb P(N=2)=\frac{\ln(2)^2}{2}+\ln(2)(1-\ln(2))$ so it seems a bit complicated to guess a form...
Is there a way out ?
