I just need a small clarification on the covariance definition of my textbook.
The definition is given to me as:
Am I correct to assume that in here
$\mu_1=E(Y_1)=\int \limits^\infty_{-\infty}y_1f1(y_1)dy_1$
$\mu_1=E(Y_1)=\int \limits^\infty_{-\infty}y_2f_2(y_2)dy_2$
Where:
$f_1(y_1)=\int \limits^\infty_{-\infty}f(y_1,y_2)d y_2$
$f_2(y_2)=\int \limits^\infty_{-\infty}f(y_1,y_2)d y_1$
?
You compute $\mu_1$ as $$\mu_1 = \int\limits_{- \infty}^{\infty} \int\limits_{- \infty}^{\infty}y_1 f(y_1,y_2)dy_2 dy_1$$ which is also $$\mu_1 = \int\limits_{- \infty}^{\infty} y_1 \Big( \int\limits_{- \infty}^{\infty} f(y_1,y_2)dy_2 \Big) dy_1$$ If you call $f_1(y_1) =\int\limits_{- \infty}^{\infty} f(y_1,y_2)dy_2 $, then $$\mu_1 = \int\limits_{- \infty}^{\infty} y_1 f_1(y_1) dy_1$$. So what you mention is correct. Similarly, you apply the same for $\mu_2$