Two standard normal variables $Z_1$ and $Z_2$ have covariance 0.3. Find the mean and variance of the random variable $X = 3Z_1- Z_2$. I found out the variance by using the formulae $$\textrm{Var}[X+Y]=\textrm{Var}[X]+ 2\textrm{cov}(X,Y) +\textrm{Var}[Y]$$ however I am not able to find out the mean (which i think is the expected value).
In the second question i also have to find the PDF when $Z_1$ and $Z_2$ are independent. Seems like i am missing a trigger point for the mean. Can someone route me to the right direction?
Yes, the mean is the expected value or expectation and is a linear function, which means that for any numbers $a,b$ and any random variables $Z_1,Z_2$ (dependent, correlated or not) $$\mathbb E[aZ_1+bZ_2]=a\mathbb E[Z_1]+b\mathbb E[Z_2]$$ so actually this was the easy part. For the second part, if $Z_1,Z_2$ are independent, then $X$ follows again the normal distribution with parameters the ones you specified above.
\begin{align}\text{Var}(X)&=3^2\text{Var}(Z_1)+\text{Var}(Z_2)-2\cdot3\cdot\text{cov}(Z_1,Z_2)=9\cdot1+1-6\cdot0.3=8.2\\[0.2cm]\mathbb E[X]&=3\mathbb E[Z_1]-\mathbb E[Z_2]=3\cdot0-0=0\end{align}