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A doubt on stastical independence , orthogonality and uncorrelatedness ?

Hi friends,

I wanted to make my concepts on statistical independence, uncorrelatedness and orthogonality clear. Suppose I have 2 random variables x and y. I have 2 pictures on the above concepts, which is more general picture? If you finds any mistake in the picture , please point it out.

Attachment 25293

Picture (a)

Attachment 25294

Picture (b)

What is statistical independence and linear independent independence means? Are they same?

Do pdf(x,y)=pdf(x)*pdf(y) always imply E(XY)=E(X)E(Y). Can any one please explain that?

-Devanand T

Re: A doubt on stastical independence , orthogonality and uncorrelatedness ?

Hey dexterdev.

The second diagram is a better representation.

Independent will always give you E[XY] = E[X]E[Y] and there can be overlap in orthogonality with independence. The orthogonal ones should always have the property of being un-correlated though (the covariance matrix should be diagonal for this case).

Re: A doubt on stastical independence , orthogonality and uncorrelatedness ?

Hello Chiro,

Are statistical independence and linear independent independence same ?

How do pdf(x,y)=pdf(x)*pdf(y) always imply E(XY)=E(X)E(Y) ?

-Thanks

Re: A doubt on stastical independence , orthogonality and uncorrelatedness ?

Yes they are the same.

For independence you have P(X = x, Y = y) = P(X = x)*P(Y=y).

I thought about doing a derivation but the easiest way to prove this is to use what is known as Fubini's Theorem:

Fubini's theorem - Wikipedia, the free encyclopedia

Basically if you look at the corollary, just replace g(x) with x*P(X=x) and h(y) with y*P(Y=y) and thats the proof done.

Re: A doubt on stastical independence , orthogonality and uncorrelatedness ?

Re: A doubt on stastical independence , orthogonality and uncorrelatedness ?

Quote:

Originally Posted by

**chiro** Hey dexterdev.

The second diagram is a better representation.

Independent will always give you E[XY] = E[X]E[Y] and there can be overlap in orthogonality with independence. The orthogonal ones should always have the property of being un-correlated though (the covariance matrix should be diagonal for this case).

Then what does this pdf says : (it has a different picture with linear independence etc)

http://www.psych.umn.edu/faculty/wal...gs/rodgers.pdf

Re: A doubt on stastical independence , orthogonality and uncorrelatedness ?

The above refers to observations: I was referring above to probabilistic independence (in terms of the independence of random variables through probabilistic properties) which is a little different.