Expectation of Binomial Distribution/Proof 4
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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 Moment Generating Function of Binomial Distribution, the moment generating function of $X$, $M_X$, is given by:
- $\ds \map {M_X} t = \paren {1 - p + p e^t}^n$
By Moment in terms of Moment Generating Function:
- $\ds \expect X = \map {M_X'} 0$
We have:
\(\ds \map {M_X'} t\) | \(=\) | \(\ds \frac \d {\d t} \paren {1 - p + p e^t}^n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {\frac \d {\d t} } {1 - p + p e^t} \map {\frac \d {\map \d {1 - p + p e^t} } } {1 - p + p e^t}^n\) | Chain Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds n p e^t \paren {1 - p + p e^t}^{n - 1}\) | Derivative of Exponential Function, Derivative of Power |
Setting $t = 0$ gives:
\(\ds \expect X\) | \(=\) | \(\ds n p e^0 \paren {1 - p + p e^0}^{n - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds n p \paren {1 - p + p}^{n - 1}\) | Exponential of Zero | |||||||||||
\(\ds \) | \(=\) | \(\ds n p\) |
$\blacksquare$