 Solution

The event $Y_n\le z$ occurs precisely when every one of the first $n$ levels is at most $z$. Their independence gives the cumulative distribution functionFor any nonnegative random variable, the pointwise identity $Y_n={\int }_0^{\infty }{1}_{\{Y_n>z\}}dz$ and the Tonelli theorem allow interchange of expectation and integral. HenceThis identity holds as an extended nonnegative integral; finiteness of the mean is not guaranteed by continuity of $F$.