This [[Random Variable]] can only take outcomes from $\{0,1\}$. It typically models success or failure. $ X= \begin{cases} 1 & \text{w.p.} &p\\ 0 & \text{w.p.} &(1-p)\\ \end{cases} $ **Indicator variable:** To translate an [[Events|event]] $A$ into a r.v., we use a Bernoulli to denote if the event happened or not. $ I_A= \begin{cases} A & \text{w.p.} &p\\ A^C & \text{w.p.} &(1-p)\\ \end{cases} \implies \mathbf P(I_A)=\mathbf P(A) $ **Expectation:** $ \mathbb E[X] = 1_p +0_(1-p)=p $ **Variance:** $ \mathrm{Var}(X) = p*(1-p) $ ![[bernoulli-variance.png|center|400]] Variance derivation via expected value rule: $ \begin{align} \mathrm{Var}(X) &=\sum_x (x-\mu)^2*p_X(x) \\ &=(1-p)^2*p + (0-p)^2*(1-p) \\[4pt] &=(1-2p+p^2)*p+p^2*(1-p) \\[4pt] &=p-2p^2+p^3+ p^2-p^3 \\[4pt] &=p-p^2 \\[4pt] &=p*(1-p) \end{align} $ Variance derivation via method of moments: $ \begin{align} \mathrm{Var}(X) &=\mathbb E[X^2]- (\mathbb E[X])^2 \\ &=\mathbb E[X]- (\mathbb E[X])^2 \\ &=p-p^2 \\ &=p*(1-p) \end{align} $ where: - (2) For a Bernoulli r.v. its second moment is equal to its first moment (i.e. [[Expectation]]). This is because Bernoulli $\in \{0,1\}$, and squaring these values results in the same outcome. - (3) Replacing the expectation with $p$.