Q:

Find the variance of z=2x-3y

Find the variance of Z = 2x-3y. Assume X and Y are independent with X as Geometric with p = 1/2 and Y as Geometric with p = 1/3.
I know how to find the variance for each Geometric distribution using the formula:
$$\sigma^{2}=\frac{1-p}{p^{2}}$$
Using the above, I got 2 for Geometric with p = 1/2 and 6 for Geometric with p = 1/3. My question is how do I use this information to find the correct value for Z?

A:

$$\mathbb{E}[Z] = 2\mathbb{E}[X] - 3\mathbb{E}[Y]$$ (linearity of expectation)
$$\mathbb{E}[Z^2] = 4\mathbb{E}[X^2] + 9\mathbb{E}[Y^2] - 12\mathbb{E}[XY] = 4\mathbb{E}[X^2] + 9\mathbb{E}[Y^2] - 12\mathbb{E}[X]\mathbb{E}[Y]$$ (independence)
You already know how to compute the mean and second moment for $X$ and $Y$. You can plug those into the above and compute $Var(Z) = \mathbb{E}[Z^2] - \mathbb{E}[Z]^2$. 

