Covariance equality for independent Gaussians

Question

$X = Y + Z$ are independent Gaussians with zero means. I'm told that:$$Cov(Y,X) = \sigma_{Y}^2$$

Could someone show me the intermediate steps to get there?

Answer 1

Because $$\text{Cov}(Y,X)=\text{Cov}(Y,Y+Z)=\text{Cov}(Y,Y)+\text{Cov}(Y,Z)=\text{Var}(Y)$$ Since $\text{Cov}(Y,Z)=0$ because of their independence.
Remember that $$\mathbb{E}\left[ \sum X_i\right]=\sum \mathbb{E}[X_i]=\sum \mu_i$$ Then $$\begin{align} \text{Cov}\left (\sum X_i, \sum Y_i\right) &= \mathbb{E}\left [\left(\sum X_i-\sum \mu_i\right)\left(\sum Y_i-\sum \nu_i\right)\right] \\ &= \mathbb{E}\left [\sum \left(X_i- \mu_i\right)\sum\left( Y_i- \nu_i\right)\right]\\ &=\mathbb{E}\left [\sum \sum \left(X_i- \mu_i\right)\left( Y_i- \nu_i\right)\right]\\ &=\sum \sum \mathbb{E}\left [\left(X_i- \mu_i\right)\left( Y_i- \nu_i\right)\right]\end{align}$$