How does the following equation evaluate?

Question

$$E[n_i^2] = E[(\sum_{j=1}^n X_{ij})^2]$$ $$= E[\sum_{j=1}^n\sum_{k=1}^n X_{ij} X_{ik}]$$

Here $E$ is expected value and $n_i$ is a random variable. $X_{ij}$ is an indicator random variable. How does the equation decompose to latter part.

Answer 1

It is just expanding the multiplication, like for example

$$ \left(\sum_{j=1}^3 x_j\right)^2 = (x_1+x_2+x_3)^2 = (x_1+x_2+x_3)(x_1+x_2+x_3) \\ = x_1 x_1 + x_1 x_2 + x_1 x_3 + x_2 x_1 + x_2 x_2 + x_2 x_3 + x_3 x_1 + x_3 x_2 + x_3 x_3\\ = \sum_{j=1}^3 \sum_{k=1}^3 x_j x_k$$