How do I calculate this expectation?

Question

This is a very trivial question but I have a lot of trouble dealing with Expectation, summation and product together. So I need find this:

$\mathbb{E}(\sum_{i=1}^n m_i X_i)$

where $m_i$ are constants.

I know that next step would be:

$\sum_{i=1}^n \mathbb{E}(m_i X_i)$

After this, would the next step be:

$\sum_{i=1}^n m_i . \mathbb{E}(X_i)$

or

$\sum_{i=1}^n m_i . \sum_{i=1}^n\mathbb{E}(X_i)$

?

Thank you.

Answer 1

Possibly the best way to think about this is to start with a simple example. If $n = 2$, then $\sum_{i = 1}^2 m_i X_i = m_1 X_1 + m_2 X_2$. Then if we look at the expectation of it, we get

$\begin{eqnarray}\mathbb{E}(\sum_{i = 1}^2 m_i X_i) & = & \mathbb{E}(m_1 X_1 + m_2 X_2) \\ & = & \mathbb{E}(m_1 X_1) + \mathbb{E}(m_2 X_2) \\ & = & m_1 \mathbb{E}(X_1) + m_2 \mathbb{E}(X_2) \end{eqnarray}$

you should be able to simplify that last line into a sigma form, and from there you can intuit what happens with more general $n$.