Let X follows gamma distribution and the PDF is given as x : \"1 F x) Bar'ax e ,x > 0,51 > 0.8 > 0. (a) ('5 points) Show that I: x\"'le_dx = B'II'a. (b) (15 points) Derive the expectation and variance of X. (c) (5 points) If at = 1, does it match with a known distribution? Name that distribution. (d) (10 points) Derive the E [6\"]. (e) (15 points) Write your derived expression from part (d) and name it m(t) = E [8\"]. Determine m'(t) = gum?) and then plug in t = 0, i.e. 171(0). Does this expression matches with the E [x]