6. (Total: 18 points) Consider a standard uniform random variable UU(0,1). (a) (3 points) Derive the moment generating function of U. (b) (3 points) Write down the mean, variance, and skewness of U. (c) (3 points) Find the kurtosis of U. (d) (9 points) Write down the distribution (with parameterization) of the following transformed random variables: (i) 1(U) where is the cumulative distribution function of the standard normal distribution (ii) lnU (iii) 1U