# Thread: Correlation between two normal variables ...

1. ## Correlation between two normal variables ...

Suppose

A ~ N (0,a),

B ~ N (0,b-a),

C = A + B

what is the expression for the correlation between A and C?

2. ## Re: Correlation between two normal variables ...

Are A and B independent? Nothing you posted has given information about the dependency between A and B.

3. ## Re: Correlation between two normal variables ...

Sorry, yes, they are independent.

4. ## Re: Correlation between two normal variables ...

Is this for homework? What have you tried so far?

5. ## Re: Correlation between two normal variables ...

I attempted to start with cov(A,C) = E(AC) - E(A)E(C), but the problem is, I am sure the answer is not zero, but I am unsure how to show that. Intuitively when a = b, corr = 1 and when b --> inf corr --> 0.

6. ## Re: Correlation between two normal variables ...

Originally Posted by BigBugBuzzz
I attempted to start with cov(A,C) = E(AC) - E(A)E(C), but the problem is, I am sure the answer is not zero, but I am unsure how to show that. Intuitively when a = b, corr = 1 and when b --> inf corr --> 0.
The correlation is not zero. And, I think you should define your variables as, say, X, Y, and Z so you don't get confused. Anyway, just use the definition of the correlation coefficient and the properties of expectation.

Corr = Var[X] / ( Sqrt[ Var[X] + Var[Y - X] ] * Std[X])

7. ## Re: Correlation between two normal variables ...

The following does not appear right...

First, let A = X and C = Y as suggested.

X ~ N (0,a)

Y = X + N ~ (0,b-a) = N ~ (X,b-a)

Now applying the formula given:

Corr = Var[X] / ( Sqrt[ Var[X] + Var[Y - X] ] * Std[X]) -->

Corr = a / ( sqrt(a + b) * sqrt(a) )

I would have thought that when a = b, corr = 1, but that does not follow from my attempt...

8. ## Re: Correlation between two normal variables ...

What did you put as Var[Y - X]? It doesn't look to me that you put the right quantity there.

9. ## Re: Correlation between two normal variables ...

Var[Y - X] = b-a + a

That is what I suspected was wrong. Do I not add the variances, even when subtracting two normal variables?

10. ## Re: Correlation between two normal variables ...

In terms of the constants, I think you should end up with a correlation of:

11. ## Re: Correlation between two normal variables ...

I don't think that's quite right. I get r = a/sqrt(ab) as the final answer.

12. ## The Following User Says Thank You to Dason For This Useful Post:

BigBugBuzzz (02-04-2012)

13. ## Re: Correlation between two normal variables ...

X ~ N(0, a)
Y ~ N(0, b - a)
Z = X + Y

I used Cov(X, X+Y) = Cov(X, X) + Cov(X, Y) = Var(X) + Cov(X, Y) = Var(X) by independence. Cor(X, Z) = Cov(X,Z)/sqrt(Var(X)Var(Z)).

14. ## Re: Correlation between two normal variables ...

Dason, I just checked the equation I wrote empirically and it appears correct.

In terms of the Var[z], did you use (b-a)^2 when you substituted?

15. ## Re: Correlation between two normal variables ...

Var(Z) = b. Since X and Y are independent Var(X + Y) = Var(X) + Var(Y) = a + (b - a) = b.

16. ## Re: Correlation between two normal variables ...

Originally Posted by Dason
Var(Z) = b. Since X and Y are independent Var(X + Y) = Var(X) + Var(Y) = a + (b - a) = b.
Ah, I think I see the problem now....you're treating a and b as variances and not standard deviations.

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