Distribution until the First Increase?

Hey,

On a homework I've been given this problem. I don't even know where to start. Is there a name for this distribution or a way I can look up more information about it? Thanks!

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Let $\displaystyle X_1, X_2,...$ be independent identically distributed continous random variables. We call $\displaystyle n$ the first time of increase if

$\displaystyle X_1>X_2>X_3>...>X_{n-1}<X_n$

Let $\displaystyle N$ be the time until the first increase. Show that $\displaystyle E[N]=e$.

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The hint we were given was to use the tail sum formula for a random variable taking values in the positive integers. But how do you go from a discrete formula to a continuous one? I am completely lost. What is this distribution called?

Thanks guys!