Help Determing What Distribution

#1
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I believe I have answered a), as L~Gamma(N,1/20), therefore mgf = (20/(20-t))^N (if wrong please feel free to correct me). It has been defined that L=-ln(U), and I am finding difficulty in determing what distribution U would be classified as. Once classfieid, I believe that finding the mean, standard deviation, and cumulative probability will be simple, but I am stuck. Any help appreciated