Editing Exponential distribution (section)

===Bayesian inference===
The [[conjugate prior]] for the exponential distribution is the [[gamma distribution]] (of which the exponential distribution is a special case).  The following parameterization of the gamma probability density function is useful:

<math display="block">\operatorname{Gamma}(\lambda; \alpha, \beta) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} \lambda^{\alpha-1} \exp(-\lambda\beta).</math>

The [[posterior distribution]] ''p'' can then be expressed in terms of the likelihood function defined above and a gamma prior:

<math display="block">\begin{align}
  p(\lambda) &\propto L(\lambda) \Gamma(\lambda; \alpha, \beta) \\
    &= \lambda^n \exp\left(-\lambda n\overline{x}\right) \frac{\beta^{\alpha}}{\Gamma(\alpha)} \lambda^{\alpha-1} \exp(-\lambda \beta) \\
    &\propto \lambda^{(\alpha+n)-1} \exp(-\lambda \left(\beta + n\overline{x}\right)).
\end{align}</math>

Now the posterior density ''p'' has been specified up to a missing normalizing constant.  Since it has the form of a gamma pdf, this can easily be filled in, and one obtains:

<math display="block">p(\lambda) = \operatorname{Gamma}(\lambda; \alpha + n, \beta + n\overline{x}).</math>

Here the [[Hyperparameter (Bayesian statistics)|hyperparameter]] ''α'' can be interpreted as the number of prior observations, and ''β'' as the sum of the prior observations.
The posterior mean here is:
<math display="block">\frac{\alpha + n}{\beta + n\overline{x}}.</math>