Linear combination of gaussian variables

Question

If $X\sim N(0,\sigma_1^2)$,$Y\sim N(0,\sigma_2^2)$ and given that X,Y are independent random variable with normal distributions, then for the random variable $U=\alpha X+\beta Y\sim N(\mu,\sigma^2)$ ($\alpha,\beta$ are constants), What is the mean and variance of this new Gasusian distribtuion $U$ in closed form? Does this generalise indefinitely? Is $\mu=0$ and $\sigma^2=\alpha^2\sigma_1^2+\beta\sigma_2^2$?

Answer 1

So, close. Indeed $\mu = 0$, but $\sigma = \sqrt{\alpha^2 \sigma_1^2 + \beta^2 \sigma_2^2} $.

I suspect you just mis-typed leaving out the power of 2 in both sigma and beta.

And yes it does generalize.