I am trying to predict the covariance of two linear combinations of normal random variables:

$X = wN(u_1,\sigma^2_1)+(1-w)N(u_2,\sigma^2_2)$
$Y = wN(u_1,\sigma^2_1)+(1-w)N(u_3,\sigma^2_3)$

where $w$ can range from 0 to 1.

I've tried solving for $\text{cov}(X,Y)$ using

$\text{cov}(X,Y) = \text{E}(XY) - \text{E}(X)\text{E}(Y) \\ \text{cov}(X,Y) = \text{corr}(X,Y)\sigma_X\sigma_Y$

but am not sure how to find $\text{E}(XY)$ in the first case and $\text{corr}(X,Y)$ in the second.

Any pointers would be greatly appreciated.

I am trying to predict the covariance of two linear combinations of normal random variables: $\newcommand{\N}{\mathcal N}$ \begin{align} X &= w\N(u_1,\sigma^2_1)+(1-w)\N(u_2,\sigma^2_2) \\ Y &= w\N(u_1,\sigma^2_1)+(1-w)\N(u_3,\sigma^2_3) \end{align} where $w$ can range from $0$ to $1$.

I've tried solving for $\text{cov}(X,Y)$ using \begin{align} \text{cov}(X,Y) &= \text{E}(XY) - \text{E}(X)\text{E}(Y) \\ \text{cov}(X,Y) &= \text{corr}(X,Y)\sigma_X\sigma_Y \end{align}

but am not sure how to find $\text{E}(XY)$ in the first case and $\text{corr}(X,Y)$ in the second.

I am trying to predict the covariance of two linear combinations of normal random variables:

$X = wN(u_1,\sigma^2_1)+(1-w)N(u_2,\sigma^2_2)$
$Y = wN(u_1,\sigma^2_1)+(1-w)N(u_3,\sigma^2_3)$

where $w$ can range from 0 to 1.

I've tried solving for $cov(X,Y)$ using

$cov(X,Y) = E(XY) - E(X)E(Y) \\ cov(X,Y) = corr(X,Y)\sigma_X\sigma_Y$

but am not sure how to find $E(XY)$ in the first case and $corr(X,Y)$ in the second.

Any pointers would be greatly appreciated.

I am trying to predict the covariance of two linear combinations of normal random variables:

$X = wN(u_1,\sigma^2_1)+(1-w)N(u_2,\sigma^2_2)$
$Y = wN(u_1,\sigma^2_1)+(1-w)N(u_3,\sigma^2_3)$

where $w$ can range from 0 to 1.

I've tried solving for $\text{cov}(X,Y)$ using

$\text{cov}(X,Y) = \text{E}(XY) - \text{E}(X)\text{E}(Y) \\ \text{cov}(X,Y) = \text{corr}(X,Y)\sigma_X\sigma_Y$

but am not sure how to find $\text{E}(XY)$ in the first case and $\text{corr}(X,Y)$ in the second.

Any pointers would be greatly appreciated.