Probability : joint density function of 3 Normal Distributions

Join the discussion
Registration is free. Ask a follow-up in this thread, or start your own.
Sign up Log in
3 replies · 8K views
kkirtac
Messages
3
Reaction score
0
X1, X2, X3 are independent gaussian random variables.
Y1 = X1+X2+X3
Y2 = X1-X2
Y3 = X2-X3
are given. What is the joint pdf of Y1,Y2 and Y3 ?
 
Physics news on Phys.org
What are you confused about in this problem?

You have to show what work you've done in order to get help. And if you've made no progress whatesoever, just write what you don't understand about the problem.
 
kkirtac said:
X1, X2, X3 are independent gaussian random variables.
Y1 = X1+X2+X3
Y2 = X1-X2
Y3 = X2-X3
are given. What is the joint pdf of Y1,Y2 and Y3 ?

There seems to be some info missing here. Are X1, X2, and X3 independent? Are they discrete or continuous? Either way, what values can X1, X2, and X3 take? All this information is necessary in order to find the joint pdf of Y1, Y2, and Y3.
 
Dear,
If you are always watching the post.
the pdf of Y1 is the convolution of the three pdfs of the three random variables (X1, X2 and X3).
for Y2 and Y3, if the random variables are gaussian and centered (mean = 0) then pdf(X) = pdf(-X) and thus for pdf(Y2) = convolution of pdf(X1) and pdf(X2) while pdf(Y3) = convolution of pdf(X2) and pdf(X3).
Actually, I address you to the great book of Papoulis where you can find (for sure) the answer to your wondering .
Cheers
Manar