How to compute the expectation of an element within a vector?

Question

Say we have $\Theta = (\Theta_1, \ldots, \Theta_n)$ and $p_{\Theta}(\theta)$, how do we compute $\mathbb{E}[\Theta_j]$? Do we need to do an integral over all $\Theta_i$? If so, I don't see why. In this video, I don't understand why this happens.

Answer 1

It might help to see this in two dimensions. Let $(X,Y)$ be a random vector with joint pdf $f_{X,Y}(x,y)$ and support $A \subset \mathbb{R}^2$. Then, by definition,

$$ E[X] = \iint_{A}xf_{X,Y}(x,y)dxdy\;\;\; \& \;\;\; E[Y] = \iint_{A}yf_{X,Y}(x,y)dxdy$$

Can you try to extend this to some vector $(\Theta_1,...,\Theta_n) \in \mathbb{R}^n$? It would involve an $n$-dimensional integral.