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:
- $\ds X = \sum_{i \mathop = 1}^n Y_i$
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 |
$\blacksquare$