**1) Distribution Setup** $X \sim \text{Bern}(p), \quad p \in [0, 1]$ The Probability Mass Function (PMF) table: $\begin{array}{c|c|c} X & 0 & 1 \\ \hline P(X=x) & 1-p & p \end{array}$ Written as a function: $f_p(x) = f(x|p) = \begin{cases} p, & x=1 \\ 1-p, & x=0 \end{cases} \quad \text{for } x \in \{0, 1\}$ **2) Likelihood Function** $f(X|p) = \begin{cases} p, & X=1 \\ 1-p, & X=0 \end{cases}, \quad X \sim \text{Bern}(p)$ **3) Log-Likelihood** $\ln f(X|p) = \begin{cases} \ln p, & X=1 \\ \ln(1-p), & X=0 \end{cases}$ **4) First Derivative (Score Function)** $(\ln f(X|p))'_p = \begin{cases} 1/p, & X=1 \\ -\frac{1}{1-p}, & X=0 \end{cases}$ **5) Second Derivative** $(\ln f(X|p))''_p = \begin{cases} -1/p^2, & X=1 \\ -\frac{1}{(1-p)^2}, & X=0 \end{cases}$ $\left(-\frac{1}{1-p}\right)' = \frac{1}{(1-p)^2} \cdot (-1) = -\frac{1}{(1-p)^2}$ **6) Expected Value and Fisher Information** Summary table for expectation: $\begin{array}{c|c|c} (\ln f(X|p))'' & -\frac{1}{p^2} & -\frac{1}{(1-p)^2} \\ \hline P(X=x) & p & 1-p \end{array}$ Calculating $I(p)$: $I(p) = -\mathbb{E} [(\ln f(X|p))''] = -\left( \left(-\frac{1}{p^2}\right) \cdot p + \left(-\frac{1}{(1-p)^2}\right) \cdot (1-p) \right)$ $I(p) = -\left( -\frac{1}{p} - \frac{1}{1-p} \right)$ $I(p) = \frac{1}{p} + \frac{1}{1-p}$ $I(p) = \frac{(1-p) + p}{p(1-p)} = \frac{1}{p(1-p)}$