Question

(3 points) The random variable X has moment generating function px(t) = (0.55e +1 – 0.55) Provide answers to the following to two decimal places (a) Evaluate the natural logarithm of the moment generating function of 4X at the point t = 0.11. (b) Hence (or otherwise) find the expectation of 4X. (c) Evaluate the natural logarithm of the moment generating function of 4X + 8 at the point t = 0.11. Note: You can earn partial credit on this problem.

Answer 1

the random variable x has mgf 8(t) = (0.55 02 +0.455 Hences xn 615,0.55) @ 72 () = Elext] ext 6.55 (6455x = (o.5s (48) * (6.45 - = (0.45 + 0.5544 log out) = 5 log (0.45 + 0.55ed) etres log 8 (0.11 ) = 5 log (0.45 +0.55e94) Given t= o.ll 0.441 © =1.33 E(4x) = 4 EB) = 4*5*0.55 Hences expectation of 4x is ll.

texto © get) = Elelex: 938] - Caxton (6.55 (6.45 el & (6.55 642 6.45 x =&al (0.45 + 0.55 et 5 logre = @4+5log (045+0.5560 les toll etren, wg 8 (o.ll) = 2.21 luato