Expectation of Binomial Distribution/Proof 2

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 Bernoulli Process as Binomial Distribution, we see that $X$ as defined here is a sum of discrete random variables $Y_i$ that model the Bernoulli distribution:


 * $\displaystyle X = \sum_{i \mathop = 1}^n Y_i$

Each of the Bernoulli trials is independent of each other, by definition of a Bernoulli process. It follows that: