Expectation of Binomial Distribution/Proof 4

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

\(\ds \)

\(=\)

\(\ds n p\)

$\blacksquare$