Are A and B independent? Nothing you posted has given information about the dependency between A and B.
Suppose
A ~ N (0,a),
B ~ N (0,b-a),
C = A + B
what is the expression for the correlation between A and C?
Last edited by BigBugBuzzz; 02-03-2012 at 12:43 PM.
Are A and B independent? Nothing you posted has given information about the dependency between A and B.
Sorry, yes, they are independent.
Is this for homework? What have you tried so far?
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])
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...
What did you put as Var[Y - X]? It doesn't look to me that you put the right quantity there.
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?
I don't think that's quite right. I get r = a/sqrt(ab) as the final answer.
BigBugBuzzz (02-04-2012)
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)).
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?
Var(Z) = b. Since X and Y are independent Var(X + Y) = Var(X) + Var(Y) = a + (b - a) = b.
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