Expectation of Binomial Distribution/Proof 2

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:

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

Each of the Bernoulli trials is independent of each other, by definition of a Bernoulli process.

It follows that:

\(\ds \expect X\)

\(=\)

\(\ds \expect {\sum_{i \mathop = 1}^n Y_i }\)

\(\ds \)

\(=\)

\(\ds \sum_{i \mathop = 1}^n \expect {Y_i}\)

Sum of Expectations of Independent Trials‎

\(\ds \)

\(=\)

\(\ds \sum_{i \mathop = 1}^n p\)

Expectation of Bernoulli Distribution

\(\ds \)

\(=\)

\(\ds n p\)

Sum of Identical Terms

$\blacksquare$