expectation of $ \left(\sum_{i=1}^n {x_i} \right)^2 $

In what follows, $x_j$ are assumed to be independent.

$$ E[x_0^2] = \int_0^{\infty} \lambda t^2 e^{-\lambda t} dt = \frac{2}{\lambda^2} $$

$$ E[x_0 x_1] = \int_0^{\infty} \int_0^{\infty} \lambda^2 t_0 t_1 e^{-\lambda t_0} e^{-\lambda t_1 } dt_0 dt_1 = \frac{1}{\lambda^2} $$

$$ E[ (\sum_{j=0}^{n-1} x_j)^2 ] = n E[ x_0^2] + n(n-1) E[x_0 x_1] = \frac{ n^2 + n }{ \lambda^2 } $$