ThmDex – An index of mathematical definitions, results, and conjectures.

Result R2484 on D207: Bernoulli random boolean number

Moments of a Bernoulli random boolean number

Formulation 0

Let $X \in \text{Bernoulli}(\theta)$ be a D207: Bernoulli random boolean number.
Let $n \in \{ 1, 2, 3, \ldots \}$ be a D5094: Positive integer.

Then \begin{equation} \mathbb{E} X^n = \theta \end{equation}

Proofs

Proof 0

Let $X \in \text{Bernoulli}(\theta)$ be a D207: Bernoulli random boolean number.
Let $n \in \{ 1, 2, 3, \ldots \}$ be a D5094: Positive integer.

Using R2482: Expectation of discrete random variable, one has \begin{equation} \begin{split} \mathbb{E} X^n & = 1^n \cdot \mathbb{P}(X = 1) + 0^n \cdot \mathbb{P}(X = 0) \\ & = 1^n \theta \\ & = \theta \end{split} \end{equation} $\square$