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I have a question about the following probability:

$Pr\{\frac{\sum^N_{k=1}u_k}{N}<1\}$

where $u_k\sim \exp(1)$ are i.i.d. exponential random variables with mean one (also, $\frac{\sum^N_{k=1}u_k}{N}$ is gamma distributed).

I have plotted this probability for different $N$. The plot shows that as $N$ increases, this probability approaches $0.5$. Is this a well-known result? Has someone proved it already? If not, how to prove it rigorously?

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