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In the following problems, please follow the instructions. If you are asked to find the distribution of a random variable, first find the cumulative distribution function (cdf) of the random variable, then use the cdf to find the probability density function (pdf) of that random variable. = 3. Let R1, R2, R3 be independently distributed random variables with a common exponential distribution with mean 1. Let S = R1 + R2 + R3, and L = k(s), where k(s) is as in 2. (i) (10 points) Find the moment generating function (mgf) of S. Name the distribution with appropriate parameters. (ii) [10 points] Let R = 1/S. Find the pdf of R. (iii) (10 points] We defined the random variable L above. What are the possible values of L? Using (i) find the cdf of Land the pdf fL(1) of L (you will not have a nice, analytical expression of the pdf fL()). (iv) (10 points] Simulate 10000 values of S from the distribution you derived in (i). Using these values find the relative frequency histogram of L. Use the pdf of L that you obtained in part (iii) to numerically evaluate fl(l) for various values of I e(0, k*) (take at least 200 values). Plot the function fl(l) overlaying the relative frequency histogram of L