Variance of Poisson Distribution/Proof 1

Theorem
Let $X$ be a discrete random variable with the Poisson distribution with parameter $\lambda$.

Then the variance of $X$ is given by:
 * $\operatorname {var}\, \left({X}\right) = \lambda$

Proof 1
From the definition of Variance as Expectation of Square minus Square of Expectation:
 * $\operatorname {var}\, \left({X}\right) = E \left({X^2}\right) - \left({E \left({X}\right)}\right)^2$

From Expectation of Function of Discrete Random Variable:
 * $\displaystyle E \left({X^2}\right) = \sum_{x \mathop \in \Omega_X} x^2 \Pr \left({X = x}\right)$

So:

Then: