Paper 2, Section II, F

Probability | Part IA, 2012

(i) Define the moment generating function MX(t)M_{X}(t) of a random variable XX. If X,YX, Y are independent and a,bRa, b \in \mathbb{R}, show that the moment generating function of Z=aX+bYZ=a X+b Y is MX(at)MY(bt)M_{X}(a t) M_{Y}(b t).

(ii) Assume T>0T>0, and MX(t)<M_{X}(t)<\infty for t<T|t|<T. Explain the expansion

MX(t)=1+μt+12s2t2+o(t2)M_{X}(t)=1+\mu t+\frac{1}{2} s^{2} t^{2}+\mathrm{o}\left(t^{2}\right)

where μ=E(X)\mu=E(X) and s2=E(X2).s^{2}=E\left(X^{2}\right) . \quad [You may assume the validity of interchanging expectation and differentiation.]

(iii) Let X,YX, Y be independent, identically distributed random variables with mean 0 and variance 1 , and assume their moment generating function MM satisfies the condition of part (ii) with T=T=\infty.

Suppose that X+YX+Y and XYX-Y are independent. Show that M(2t)=M(t)3M(t)M(2 t)=M(t)^{3} M(-t), and deduce that ψ(t)=M(t)/M(t)\psi(t)=M(t) / M(-t) satisfies ψ(t)=ψ(t/2)2\psi(t)=\psi(t / 2)^{2}.

Show that ψ(h)=1+o(h2)\psi(h)=1+\mathrm{o}\left(h^{2}\right) as h0h \rightarrow 0, and deduce that ψ(t)=1\psi(t)=1 for all tt.

Show that XX and YY are normally distributed.

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