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In an electrical circuit I wish to find the noise correlation between two signal paths (which in the end are added). Let me also say, that I am quite useless at probability theory (but trying to become better!). I setup my problem as follows:

Given $X = X_1 + c_1\cdot Z$ and $Y = X_2 + c_2 \cdot Z$, where $X_1 \sim \mathcal{N}(0,\sigma^2 _1)$, $X_2 \sim \mathcal{N}(0,\sigma^2 _2)$, and $Z \sim \mathcal{N}(0,\sigma^2 _Z)$. $X$, $Y$, and $Z$ are not correlated (independent). Further, $c_1\in \mathbb{C}$ and $c_2\in \mathbb{C}$. What is the correlation coefficient between $X$ and $Y$?

As far as I have understood,

$corr(X,Y)=\frac{cov(X,Y)}{(\sigma^2_X+\sigma^2_Y)^{0.5}}$,

where $\sigma^2_X=\sigma^2_1+c_1^2\sigma^2_Z$ and $\sigma^2_Y=\sigma^2_2+c_2^2\sigma^2_Z$ - is this true with complex constants? The covariance is defined by

$cov(X,Y)=E[XY]-E[X]E[Y]$,

where $E[X]=0$ and $E[Y]=0$. How do I find $E[XY]$?

Thanks for the help in advance.

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With the use of the following:

$Cov(X,Z+Y)=Cov(X,Z)+Cov(X,Y)$

$Cov(aX+b,Y)=aCov(X,Y)$

$Cov(Y,Y)=Var(Y)$

$Cov(X,Y)=0$ if $X,Y$ are independent

we have

$Cov(X,Y)=Cov(X_1 + c_1\cdot Z, X_2 + c_2 \cdot Z)=Cov(X_1,X_2)+Cov(X_1,c_2Z)+Cov(X_2,c_1Z)+Cov(c_1Z,c_2Z)=c_1\cdot c_2 Cov(Z,Z)=c_1\cdot c_2 Var(Z)=c_1\cdot c_2 \cdot \sigma_z^2$

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