How to show that a quantity is standard normal?

Question

I'm trying to show that there exists a quantity that is a linear transformation of Gaussian random variables that is also Gaussian, any examples of this? How would I go about proving that something is Gaussian?

Answer 1

False in general. Take $Y=X$ for a counter-example.

But the result is true if $X$ and $Y$ are independent. Use characteristic functions: $Ee^{it(aX+\sqrt {1-a^{2}}Y)}=Ee^{it(aX)}Ee^{it\sqrt {1-a^{2}}Y}=e^{-a^{2}t^{2}/2}e^{-(1-a^{2})t^{2}/2}=e^{-t^{2}/2}$.

Answer 2

As pointed out, using the characteristic function is a powerful way to show this.

While it is well-known that independent Gaussian variables are stable under linear transformations, the characteristic function provides a complete description of the variable, and will often be a very reliable way to work with more complicated transformations.