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.
Relationship between variances in perfect correlation
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statistics
correlation