Expectation of Bernoulli Distribution/Proof 4

Theorem

Let $X$ be a discrete random variable with a Bernoulli distribution with parameter $p$.

Then the expectation of $X$ is given by:

$\expect X = p$

Proof

From Moment Generating Function of Bernoulli Distribution, the moment generating function of $X$, $M_X$, is given by:

$\map {M_X} t = q + p e^t$

where $q = 1 - p$.

By Moment in terms of Moment Generating Function:

$\expect X = \map {M_X'} 0$

We have:

\(\ds \map {M_X'} t\)

\(=\)

\(\ds \frac \d {\d t} \paren {q + p e^t}\)

\(\ds \)

\(=\)

\(\ds p e^t\)

Derivative of Constant, Derivative of Exponential Function

Setting $t = 0$ gives:

\(\ds \expect X\)

\(=\)

\(\ds p e^0\)

\(\ds \)

\(=\)

\(\ds p\)

Exponential of Zero

$\blacksquare$