I have tried using mgf/cdf/convolution methods to proof part ii and iii. However, I always get stuck at the end. I hope someone can guide me on how to show the proofs. Any help would be greatly appreciated. Question 2 Let X and Y be independent positive random variables. We are interested to ?nd adecreasing function T : (0, 00) > (0, 00) such that T(X +Y) is independent of T(X) T(X +Y).Such a function indeed exists with X being a Generalised Inverse Gaussian (GIG) distributionand Y a gamma distribution and T(:L’) : 1/1; This property is called Mateumoto- Y0?" propertyin the literature. Denote the density of GIGQL, a, 3)) random variable by 1 pl (a2:z:_1+b2:r)/2 f(x;p,a,b)=mx e pER, a,b>0, :L’>0, where K (p, a, b) is a constant depending only on p, a, b. Denote the density of a gamma random variable 70¢, a) bya gem», 0») = MW18? a. a > 0, where Fur) is the Gamma function. Let T($) : 1/1:, x > O. (i)  Let X be a GIGQL, a, b) random variable. Show that T(X) is distributed as GIG(,u, b, (1.). (ii)  If X N GIG(), (1,0,) and Y N "YUM (12/ 2) are independent random variables, show thatX and T(X + Y) have the same distribution. (iii)  Let X and Y be two independent random variables such that X N GIG(,u, a, b) andY N 70;,52/2), mm?) > 0. Show that T(X + Y) is independent of T(X) T(X + Y).Identify the distributions of T(X) and T(X) T(X + Y).
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