How to find correlation when two random variable W1 =4X + 3Y and W2 = 4X + 9Y are given as a linear equation of x and y.

Question

How to find correlation when two random variable $W_1 =4X + 3Y$ and $W_2 = 4X + 9Y$ are given as a linear equation of $X$ and $Y$.

Answer 1

$$\rho(W_1,W_2) = \frac{Cov(W_1,W_2)}{\sqrt{Var(W_1)V(W_2}}$$

The covariance is a bilinear function, so :

$Cov(W_1,W_2) = Cov(4X+3Y, 4X+9Y) = 16 Var(X) + 27 Var(Y) + 48 Cov(X,Y)$

Assuming that $X$ and $Y$ are independent, we have $Cov(X,Y)=0$ and $Var(W_i) = a_i^2 Var(X) + b_i^2 Var(Y)$.

Finally (in the independent case) :

$$\rho(W_1,W_2) = \frac{16 Var(X) + 27 Var(Y)}{\sqrt{(16Var(X)+9Var(Y))(16Var(X)+81Var(Y))}}$$

I let you finish the simplification.

If $X$ and $Y$ are not independent, you need to calculate $Cov(X,Y)$ but also $Var(W_i) = Cov(W_i, W_i) = a_i^2 Var(X) + b_i^2 Var(Y) + 2a_ib_i Cov(X,Y)$.

I think you have everything you need to know what to do next.