Relationship between Covariance, Variance and Correlation

Question

I am studying these slides : Random Forest. Slide 6 explains how to reduce the variance and at some point in the mathematical derivation, the covariance equals the variance times correlation (line 3) can someone explain this step mathematically?

Answer 1

In general, if two variables $X,\,Y$ have standard deviations $\sigma_X,\,\sigma_Y$ and correlation coefficient $\rho$, their covariance is $\rho\sigma_X\sigma_Y$. Each $T_i$ has variance $\sigma^2$ and standard deviation $\sigma>0$, so if $T_i,\,T_j$ have correlation $\rho$ their covariance is $\rho\cdot\sigma\cdot\sigma=\sigma^2\rho$.

Answer 2

In general if $X,Y$ are non-degenerate random variables defined on the same probability space that both have a second moment then the correlation of $X$ and $Y$ is defined by:$$\rho(X,Y)=\frac{\mathsf{Cov(X,Y)}}{\sigma_X\sigma_Y}$$where $\sigma_X^2:=\mathsf{Var}(X)$ and $\sigma_Y^2:=\mathsf{Var}(Y)$.

In special case $\sigma_X=\sigma_Y=\sigma$ this relation can be written as:$$\sigma^2\rho(X,Y)=\mathsf{Cov}(X,Y)$$