Expectation of Binomial Distribution

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:
 * $$E \left({X}\right) = \sum_{x \in \operatorname{Im} \left({X}\right)} x \Pr \left({X = x}\right)$$

Thus:

$$ $$ $$ $$ $$