# How can I prove X = (X1, X2, X3) is independent?

1. Apr 23, 2010

### kkjs

For (Y1, Y2, Y3, Y4) ~ D4(1,2,3,4;5)
let Xk = [∑(from i=1 to k) Yi] / [∑(from i=1 to k+1) Yi] where k = 1,2,3

How can I prove X = (X1, X2, X3) is independent?
What I did was...

(Y1, Y2, Y3, Y4) ~ D4(1,2,3,4;5) = (Z1, Z2, Z3, Z4) / (Z1+Z2+Z3+Z4+Z5) where Z ~ N(0,1), Z IID G(1/2)

Now, we have
X1 = Z1 / (Z1+Z2)
X2 = (Z1+Z2) / (Z1+Z2+Z3)
X3 = (Z1+Z2+Z3) / (Z1+Z2+Z3+Z4)

I think if i can somehow show X1 and X2 are independent and X2 and X3 are independent then X1 and X3 are independent as well but how??? this is a part I don't get T-T

2. Apr 23, 2010

Re: independence

I can't understand your notation, hopefully someone will jump in, but in general, if you're talking about linear independence of vectors, it's not a transitive relation.

3. Apr 23, 2010

### vela

Staff Emeritus
Re: independence

That won't work in general. What if, for example, X3 turned out to be a multiple of X1? If X1 and X2 are independent, then X2 and X3 would also be independent, but X1 and X3 are definitely not independent.

Perhaps you can make it work because of the way the X's are constructed in your specific problem, but I'd guess it's not the right strategy.