Let $X$ and $Y$ be indepedent, identically distributed standard normal random variables, i.e., the probability density function of $X$ is
\[f_{X}(x)=\frac{1}{\sqrt{2 \pi}} \exp \left(-\frac{x^{2}}{2}\right),-\infty<x<\infty. \]
The random variable $Z$ is defined as $Z=a X+b Y$, where $a$ and $b$ are non-zero real numbers. What is the probability density function of $Z$ ?
- $\frac{1}{\sqrt{2 \pi}} \exp \left(-\frac{z^{2}}{2}\right)$
- $\frac{1}{\sqrt{2 \pi\left(a^{2}+b^{2}\right)}} \exp \left(-\frac{z^{2}}{2\left(a^{2}+b^{2}\right)}\right)$
- $\frac{|z|}{2} \exp \left(-\frac{z^{2}}{2}\right)$
- $\frac{|z|}{2\left(a^{2}+b^{2}\right)} \exp \left(-\frac{z^{2}}{2\left(a^{2}+b^{2}\right)}\right)$
- none of the above