3.2 Answers

1. Suppose X is binomially distributed:

\[X\sim Bin(1,\theta) \\ f(x|\theta) = \theta^x (1-\theta)^{1-x}\]

Then, we follow the process:

\[log \;f(x|\theta) = x·log(\theta) + (1-x)·log(1-\theta)\]

\[\frac{d}{d \theta} log \;f(x|\theta) = \frac{x}{\theta} - \frac{1-x}{1-\theta}\]

\[\frac{d^2}{d\theta^2} log \;f(x|\theta) = -\frac{x}{\theta^2}-\frac{1-x}{(1-\theta)^2}\]

Because \(E(X) = \theta\), we know that

\[\begin{align} I(\theta) & = -E_{\theta} \left[ \frac{d^2}{d\theta^2} log \;f(x|\theta)\right] \\ & = \frac{\theta}{\theta^2} + \frac{1-\theta}{(1-\theta)^2} \\ & = \frac{1}{\theta(1-\theta)} \end{align}\]