Find $E[X_{(1)}X_{(2)}]$ where $X_1, X_2 \sim N(0,\sigma^2)$

Question

My goal is to find the covariance of $X_{(1)}$ and $X_{(2)}$ and I was able to figure out

$$E[X_{(1)}]=-\frac{\sigma}{\sqrt{\pi}} \quad \text{and} \quad E[X_{(2)}]=\frac{\sigma}{\sqrt{\pi}}$$

However, I am not comfortable with the expectation of the product of the order statistics.

I know that $X_1$ and $X_2$ are independent but I am sure $X_{(1)}$ and $X_{(2)}$ are dependent.

So, I am not quite sure what the joint distribution would be and hence I am stuck finding the expectation.

I would appreciate your help.

Answer 1

$X_{(1)}X_{(2)}$ is same as $X_1X_2$ so $EX_{(1)}X_{(2)}=EX_1X_2$.