For (Y(adsbygoogle = window.adsbygoogle || []).push({}); _{1}, Y_{2}, Y_{3}, Y_{4}) ~ D_{4}(1,2,3,4;5)

let X_{k}= [∑(from i=1 to k) Yi] / [∑(from i=1 to k+1) Yi] where k = 1,2,3

How can I prove X = (X_{1}, X_{2}, X_{3}) is independent?

What I did was...

(Y_{1}, Y_{2}, Y_{3}, Y_{4}) ~ D_{4}(1,2,3,4;5) = (Z_{1}, Z_{2}, Z_{3}, Z_{4}) / (Z_{1}+Z_{2}+Z_{3}+Z_{4}+Z_{5}) where Z ~ N(0,1), Z IID G(1/2)

Now, we have

X_{1}= Z_{1}/ (Z_{1}+Z_{2})

X_{2}= (Z_{1}+Z_{2}) / (Z_{1}+Z_{2}+Z_{3})

X_{3}= (Z_{1}+Z_{2}+Z_{3}) / (Z_{1}+Z_{2}+Z_{3}+Z_{4})

I think if i can somehow show X_{1}and X_{2}are independent and X_{2}and X_{3 }are independent then X_{1}and X_{3}are independent as well but how??? this is a part I don't get T-T

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

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