Expectation of Binomial Distribution/Proof 2

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Theorem

Let $X$ be a discrete random variable with the binomial distribution with parameters $n$ and $p$ for some $n \in \N$ and $0 \le p \le 1$.


Then the expectation of $X$ is given by:

$\expect X = n p$


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:

\(\displaystyle E \left({X}\right)\) \(=\) \(\displaystyle E \left({ \sum_{i \mathop = 1}^n Y_i }\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{i \mathop = 1}^n E \left({Y_i}\right)\) Sum of Expectations of Independent Trials‎
\(\displaystyle \) \(=\) \(\displaystyle \sum_{i \mathop = 1}^n \ p\) Expectation of Bernoulli Distribution
\(\displaystyle \) \(=\) \(\displaystyle n p\) Sum of Identical Terms

$\blacksquare$