Question

Let X1;X2;X3;...;X6 be a random sample from a distribution with the following probability desity function:

Fx(x) = (1+4x)/3 for 0<x<1
= 0 for x=<0; x>=1.

(a) Determine the joint p.d.f of Y1 and Y6 where Y1<Y2<...Y6 are the order statistics.
(b) Let R= Y6-Y1 and Z=Y1 in the joint p.d.f in (a).
(i) Find the inverses of the
functions for r and z.
(ii) Compute the Jacobian.
(iii) Determine the joint
probability density
function of R and Z.
(iv) Write down the integral you
would have to solve to
determine the p.d.f of R.
Do not solve the integral.