Covariance of linear combinations of correlated random variables 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. 
 A: HINT:
\begin{align*}
cov(aX+bY, cV + dW) &= E[acXV + adXW + bcYV + bd YW]\\
&-E[aX+bY]E[cV+dW]\\
&= acE[XV]+adE[XW]+ bcE[YV] + bdE[YW]\\
&- acE[X]E[V]-adE[X]E[W]-bcE[Y]E[V]-bdE[Y]E[W]\\
&= ac \times cov(X,V) + ad \times cov(X,W) + bc\times cov(Y,V) + bd \times  cov(Y,W) 
\end{align*}
A: w is a constant ranging from 0 to 1.
Let P~ N(u1,σ21), Q ~ N(u2, σ22) and R~ N(u3, σ23)]
If variables P, Q, and R are independent, then
Cov (X, Y) = Cov [w*N(u1,σ21)+(1−w)N(u2, σ22), wN(u1,σ21)+(1−w)*N(u3, σ23)]
= Cov [wP + (1-w) Q,  w*P + (1-w)*R]
= w2*Cov(P, P) + w*(1-w)Cov(P, R) + + w(1-w)*Cov(Q, P) +  (1-w)2*Cov(Q, R)
= w2* σ21  [If variables  P, Q, and R are pair-wise independent]
Alternately, by using Cov(X, Y) =E(XY)−E(X)E(Y)
Cov (X, Y) = Cov [wP + (1-w) Q,  w*P + (1-w)*R]
= E[w2*P2 + w*(1-w)PR + + w(1-w)*QP + (1-w)2*QR] – 
E[wP + (1-w) Q]* E[w*P + (1-w)*R]
= w2*E(P2) + w*(1-w)E(PR) + + w(1-w)*E(QP) + (1-w)2*E(QR)
-[w*u1 + (1-w)u2] [w*u1 + (1-w)*u3]
= w2*E(P2) - w2*u2  
[If variables  P, Q, and R are pair-wise independent, then E(XY) = E(X)*E(Y)]
= w2* σ21  
Note: if the variables P, Q and R are not independent, then we would require more information regarding the co-variance/correlation between them.
