Information in sixteen waiting times

Idea. Fisher information is minus the expected second derivative of the log-density. For the exponential distribution that second derivative does not even depend on $x$.

Step 1: one observation. $\log f(x;\lambda )=\log \lambda -\lambda x$. Then $$\frac{\partial }{\partial \lambda }\log f=\frac{1}{\lambda }-x,\frac{{\partial }^2}{\partial {\lambda }^2}\log f=-\frac{1}{{\lambda }^2}.$$ So one observation carries information $I_1(\lambda )=\frac{1}{{\lambda }^2}$.

Step 2: the sample. For independent observations the information adds: $$I_{16}(\lambda )=\frac{16}{{\lambda }^2}=\frac{16}{9}\text{at }\lambda =3.$$

What it means. By the Cramér-Rao bound, no unbiased estimator of $\lambda$ from these 16 observations has variance below $\frac{9}{16}$. The maximum likelihood estimator $\frac{16}{\sum X_i}$ is biased; the unbiased version $\frac{15}{\sum X_i}$ has variance $\frac{{\lambda }^2}{14}=\frac{9}{14}$, a little above the bound.