Raw Moment of Bernoulli Distribution/Proof 2

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Theorem

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

Let $n$ be a strictly positive integer.


Then the $n$th raw moment $\expect {X^n}$ of $X$ is given by:

$\expect {X^n} = p$


Proof

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

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

By Moment in terms of Moment Generating Function:

$\expect {X^n} = \map {M^{\paren n}_X} 0$

By Derivative of Exponential Function:

$\map {M^{\paren n}_X} t = p e^t$

Setting $t = 0$:

\(\ds \expect {X^n}\) \(=\) \(\ds p e^0\)
\(\ds \) \(=\) \(\ds p\) Exponential of Zero

$\blacksquare$