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.