Determine covariance , trouble understanding standard deviation

Question

I'm supposed to determine the covariance $C(X, Y)$ when $Y∈\mathcal{N}(-6,6) , X∈\mathcal{N}(1,10)$ and correlation coefficient: $p(X,Y) = 0.1$

By using the definition of the correlation coefficient I get:

$C(X,Y) = p(X,Y) (D(X)D(Y))$

I know that D(X) is the standard deviation which is $D(X) =\sqrt{V(X)}$

But I'm having trouble understanding the standard deviation, previously I got it from calculating the variance but now I don't understand how to do it.

Answer 1

You should rewrite a few things for clarity:

$$ X \sim \mathcal{N}(1, 10), \quad Y \sim \mathcal{N}(-6, 6), \quad \rho_{XY} = 0.1 $$

Then:

\begin{align*} \text{cov}(X,Y) &= \rho_{XY}\sigma_X\sigma_Y\\ & = \rho_{XY}\sqrt{V(X)}\sqrt{V(Y)}\\ & = 0.1\sqrt{10}\sqrt{6}\\ \end{align*}