# 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 |

$\blacksquare$