[Math] Relationship between variances in perfect correlation

Question

I have two random variables $X$ and $Y$ with mean and standard deviation $(\mu_1,\sigma_1)$ and $(\mu_2,\sigma_2)$ respectively. I know that for perfect correlation the relationship is given by a linear regression. I also know that positive perfect correlation establishes that $\mu_2$ will be linear function of $\mu_1$.
But is there a relationship between variances ? More specifically given $\mu_1,\sigma_1$ and $\mu_2$, can I evaluate what is the value of $\sigma_2$ ?
You can assume normal distributions for $X$ and $Y$ if that helps.

Answer 1

If two r.v.s $X,Y$ are perfectly correlated, $Y = mX+c$ for some constants $m,c$. So, $\mathbf{var}(Y) = m^2\mathbf{var}(X)$. Also $\mu_Y = m\mu_X+c$