Probability question. F(.) is F(X)

i) [7 Marks] Let X have a U(0, 1) distribution, with cumulative distribution function F(.). Prove that Y = - In(1 - F(X) ) has an exponential distribu- tion and give its parameter. ii) [5 Marks Derive the moment generating function (mef) of Y. iii) [4 Marks] Use the mgf of part ii) to find the distribution of Y1 + Y2 + Y3 where Y1, Y2, and Y3 are independently and identically distributed random variables with the same distribution as Y of part i) and ii). What is the name of this distribution