Expectation of Binomial Distribution/Proof 1

Theorem
Let $X$ be a discrete random variable with the binomial distribution with parameters $n$ and $p$.

Then the expectation of $X$ is given by:
 * $E \left({X}\right) = n p$

Proof
From the definition of expectation:
 * $\displaystyle E \left({X}\right) = \sum_{x \mathop \in \Omega_X} x \Pr \left({X = x}\right)$

Thus: