4 5 of 5 Question 4 4. (a) Assume T > 0, and that the moment generating function Mx (t) of a random variable X is finite for t

4

Extracted text: 5 of 5 Question 4 4. (a) Assume T > 0, and that the moment generating function Mx (t) of a random variable X is finite for t < t.="" explain="" the="" expansion="" mx="" (t)="1+" ut="" +s?t²="" +="" o(t³),="" 2="" %3|="" where="" u="E(X)" and="" s2="" e(x²).="" [you="" may="" assume="" the="" validity="" of="" interchanging="" expectation="" and="" differentiation.]="" (b)="" let="" x,="" y="" be="" independent,="" identically="" distributed="" random="" variables="" with="" mean="" 0="" and="" variance="" 1,="" and="" assume="" their="" moment="" generating="" function="" m="" satisfies="" the="" condition="" of="" part="" (a)="" with="" t="o." suppose="" that="" x="" +y="" and="" x="" -="" y="" are="" independent.="" (i)="" using="" m(2t)="" deduce="" that="" (t)="" :="M(t)/M(-t)" satisfies="" (t)="»(t/2)²." clearly="" state="" any="" property="" of="" the="" moment="" generating="" functions="" you="" use.="" (ii)="" show="" that="" b(h)="1+" o(h?)="" as="" h="" →="" 0,="" and="" deduce="" that="" (t)="1" for="" all="" t.="" (iii)="" show="" that="" x="" and="" y="" are="" normally="" distributed.="">