Question

Exercise 3. The transforms associated with two independent discrete random variables X and Y are S(e-1) P(X + Y = 15). Justify your answer. Find

Answer 1

Given that note that (e^2s/2 + e^4s/2)^7 = [e^2s times (1/2 + e^2s/2)]^7 that is (e^2s/2 + e^4s/2)^7 = e^bs times [(q + pe^as)]^n where q = p = 1/2, a = 2, b = 4 But [e^bs times (q times pe^as)]^n = e^bs times emptyset (as) where emptyset (s) = (q + pe^as)^n so the random variable associates with the moment generating functions is e^bs times emptyset (as) = az + b That is X approximately 2z + 14 where 2 approximately B (7, 1/2) similarly, e^lambda (e^s - 1) is the moment generating function associated with P(lambda) the poission random variable These, e (e^s - 1) is poission distributed with mean 8 These, y approximately p (8) (i) PX [15] = P (2z + 14 = 15) = P (2z = 1) = 0 as z can only take the values 0 to 7 only. \r\n P Y(5) = 8^5 e^- 8/5 factorial almostequalto 0.0916 E [X] = 2E [z] + 14 As z approximately B (n, p) we have E (z) = np = 7 times 1/2 = 7/2