Is this derivation of the Poisson variance correct? I mainly want to make sure I'm applying the Law of the Unconscious Statistician (LOTUS) correctly.
$ Var[X] = E[X^2] - E[X]^2 $
$ = E[X^2] - \lambda^2$
$ = \sum_{k=1}^\infty k^2\frac{\lambda^ke^{-\lambda}}{k!} - \lambda^2 \mbox{ (because of the LOTUS)}$
$ = \lambda\sum_{k=1}^\infty k\frac{\lambda^{k-1}e^{-\lambda}}{(k-1)!} - \lambda^2$
$ = \lambda\sum_{k=0}^\infty (k+1)\frac{\lambda^{k}e^{-\lambda}}{k!} - \lambda^2$
$ = \lambda E[X+1] - \lambda^2 \mbox{ (because of the LOTUS)}$
$ = \lambda (E[X]+1) - \lambda^2 = \lambda^2 + \lambda - \lambda^2 = \lambda.$