Prove that a * a standard normal + b * an independent standard normal is $\sqrt{a^{2} + b^{2}}$ * a standard normal I am learning about characteristic functions and am confused of how to show a statement in the lecture notes using characteristic functions:
a times a standard normal + b times an independent standard normal = $\sqrt{a^{2} + b^{2}}$ times a standard normal.  
I'd appreciate if someone could let me know why this can be proven using characteristic functions. 
Thanks a lot! 

Recall that the characteristic function of a sum of independent random variables is the product of the characteristic functions: writing $Ee^{itX} = \phi_X(t)$, 
$$ \phi_{aX + bY}(t) = \phi_{aX}(t) \phi_{bY}(t)$$
Recall that the characteristic function of $N(\mu,\sigma^2)$ is
$$ \phi_{N(\mu,\sigma^2)}(t)=e^{it\mu  - \frac{1}2\sigma^2 t^2}.$$
So if $X,Y$ are iid $N(0,1)$, we have $aX \sim N(0,a^2)$ and $bY \sim N(0,b^2)$, and therefore
$$ \phi_{aX + bY}(t) = e^{-\frac12(a^2 + b^2)t^2}$$
but this is the characteristic function of $N(0,a^2 + b^2)$, which means that $aX+bY\sim N(0,a^2 + b^2) = \sqrt{a^2+b^2} N(0,1)$.
