Editing Exponential distribution (section)

===Fisher information===
The [[Fisher information]], denoted <math>\mathcal{I}(\lambda)</math>, for an estimator of the rate parameter <math>\lambda</math> is given as:
<math display="block">\mathcal{I}(\lambda) = \operatorname{E} \left[\left. \left(\frac{\partial}{\partial\lambda} \log f(x;\lambda)\right)^2\right|\lambda\right] = \int \left(\frac{\partial}{\partial\lambda} \log f(x;\lambda)\right)^2 f(x; \lambda)\,dx</math>

Plugging in the distribution and solving gives:
<math display="block"> \mathcal{I}(\lambda) = \int_{0}^{\infty} \left(\frac{\partial}{\partial\lambda} \log \lambda e^{-\lambda x}\right)^2 \lambda e^{-\lambda x}\,dx = \int_{0}^{\infty} \left(\frac{1}{\lambda} - x\right)^2 \lambda e^{-\lambda x}\,dx = \lambda^{-2}.</math>

This determines the amount of information each independent sample of an exponential distribution carries about the unknown rate parameter <math>\lambda</math>.