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 $q + p = 1$.

$\blacksquare$