Question 2 (18 marks, C2) (a) Use the definition of moment generating function to find Mx(t) for the random variable X. (i) X ~ Bin(n,p) (iv) X ~ N(, ) (ii) X~ Geo(p) (iii) X (v) X~ Exp(A) For each case, state the range of the parameters and t. ~ Poi(X) (vi) X~(a, ) (b) Use moment generating function to prove the following. 2 (i) If X1 ~ Poi(\) and X2 ~ Poi(2) are independent, then X + X2 = Poi(\1 + 2). (ii) If X ~N(,) and X2 ~ N(2, 2) are independent, then 2 X + X2 = N(1 + 2,0 + 2).