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

Variance of Bernoulli random boolean number

Formulation 0

Let $X \in \text{Bernoulli}(\theta)$ be a D207: Bernoulli random boolean number.

Then \begin{equation} \text{Var} X = \theta (1 - \theta) \end{equation}

Proofs

Proof 0

Let $X \in \text{Bernoulli}(\theta)$ be a D207: Bernoulli random boolean number.

Result R2484: Moments of a Bernoulli random boolean number shows that $\mathbb{E} X = \mathbb{E} X^2 = \theta$. Using R2262: Real variance partition into first and second moments, we then have \begin{equation} \mathsf{Var} X = \mathbb{E} X^2 - (\mathbb{E} X)^2 = \theta - \theta^2 = \theta (1 - \theta) \end{equation} $\square$