Assume $Y$ is standard normal distribution on $\mathbb{R}$ and we construct a new random variable with $a,b\in \mathbb{R}^2$ such
$X=aY$
It is obvious that $X$ is not a regular normal distribution on $\mathbb{R}^2$ but on that subset $\left \{ a\cdot t\mid t\in\mathbb{R} \right \}$. My question is that how can I decide the parameters of this regular normal distribution on $\left \{ a\cdot t\mid t\in\mathbb{R} \right \}$?