MHB Correlation of Two Random Vectors

AI Thread Summary
The discussion centers on the correlation between two random vectors, X and Y, defined as X = aR + N and Y = bG + W, where R and G are strongly correlated vectors. Participants emphasize the importance of understanding the underlying concepts rather than just memorizing formulas in education. One contributor confirms that X and Y are indeed correlated, interpreting correlation in terms of expected values. The conversation highlights the need for clarity in defining correlation to arrive at accurate conclusions. Overall, the thread underscores the significance of conceptual understanding in statistical education.
OhMyMarkov
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Hello everyone!

I'm coming to notice day by day how our education is purely focused on memorizing and applying formulas rather than understanding the concept. Assume we have the following:

$X = aR + N$, and
$Y = bG + W$,

where $X, Y$ are random vectors, $R, G$ are strongly correlated random vector that average out to the zero vector each, $a, b$ are scalars, and $N, W$ are two independent vectors of i.i.d. normal RVs.

Now, $X$ and $Y$ are correlated, right?
 
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OhMyMarkov said:
Hello everyone!

I'm coming to notice day by day how our education is purely focused on memorizing and applying formulas rather than understanding the concept. Assume we have the following:

$X = aR + N$, and
$Y = bG + W$,

where $X, Y$ are random vectors, $R, G$ are strongly correlated random vector that average out to the zero vector each, $a, b$ are scalars, and $N, W$ are two independent vectors of i.i.d. normal RVs.

Now, $X$ and $Y$ are correlated, right?

I somehow suspect you have missed out some information, but under my interpretation of what you mean, yes.

If you write out what you mean by correlation it should be obvious what the answer is.

CB

PS My interpretation of what you mean when you ask are X and Y correlated is that you are asking: is \( E( (X-\overline{X}) (Y-\overline{Y})^t)\ne {\bf{0}} \)?
 
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Yes, this is what I mean. Wanted to make sure... I'll review the problem...
 
I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...

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