Quick Question: Variance of a two related Gaussian distributions

Question

I have two random variables, $X$ and $Y$. Both follow a Gaussian distribution, and $$X \sim N(0,1)\;.$$

After some manipulation, I got that $$P(X \leq z) = P(kY \leq z)$$

where $k$ is some constant.

Would this imply that $X=kY$, and therefore $\operatorname{Var}(kY)=k^2\operatorname{Var}(Y)=k^2$?

Answer 1

If $kY = X$ and $\mathsf{Var}(X) = 1$, then it must be true that $$\mathsf{Var}(kY) = 1 $$ and thus $$k^2\mathsf{Var}(Y) = 1$$ and thus $$\mathsf{Var}(Y) = \frac{1}{k^2}$$