i) [7 Marks] Let X have aU(0;1) distribution, with cumulative distribution functionF().Prove thatY=ln(1F(X)) has an exponential distribution and give its parameter.ii) [5 Marks] Derive the moment generating function (mgf) ofY.iii) [4 Marks] Use the mgf of part ii) tofind the distribution ofY1+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?

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 (mgf) of Y. iii) [4 Marks] Use the mof of part ii) to find the distribution of Y1 + Y2 + Y3 where Y1, Y2, and Y's 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