Raw Moment of Bernoulli Distribution/Proof 1

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

From the definition of expectation:

$\ds \expect {X^n} = \sum_{x \mathop \in \Img X} x^n \map \Pr {X = x}$

From the definition of the Bernoulli distribution:

$\ds \expect {X^n} = 1^n \times p + 0^n \times \paren {1 - p} = p$

$\blacksquare$