# Finding the mean and variance of a linear combination of independent random variables

If $X$ and $Y$ are independent random variables with means $\mu_x=3$ and $\mu_y=-2$ and variances $\sigma_x^2=5$ and $\sigma_y^2=3$.

How can we find the mean and variance of the random variable $Z=-X-3Y+5$?

• Hi @john. $$\color{red}{\Large\text{Welcome to MSE!}}$$ Don't worry about it this time (since you're new) but you might like to know that we prefer to use MathJax here. Also, please show your working in future. – Shaun Jan 20 '14 at 10:37

• $E(\lambda\cdot X)=\lambda\cdot E(X)$
• $E(a+X)=a+E(X)$
• $E(X+Y)=E(X)+E(Y)$
• $VAR(\lambda\cdot X)=\lambda^{2}\cdot VAR(X)$
• $VAR(a+X)=VAR(X)$
• $VAR(X+Y)=VAR(X)+VAR(Y)$