Expectation of Binomial Distribution/Proof 3
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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 the Probability Generating Function of Binomial Distribution, we have:
- $\map {\Pi_X} s = \paren {q + p s}^n$
where $q = 1 - p$.
From Expectation of Discrete Random Variable from PGF, we have:
- $\expect X = \map {\Pi'_X} 1$
We have:
\(\ds \map {\Pi'_X} s\) | \(=\) | \(\ds \map {\frac \d {\d s} } {q + p s}^n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds n p \paren {q + p s}^{n - 1}\) | Derivatives of PGF of Binomial Distribution |
Plugging in $s = 1$:
- $\map {\Pi'_X} 1 = n p \paren {q + p}$
Hence the result, as $q + p = 1$.
$\blacksquare$