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The parameter space of a statistical model is the set of allowed statistical parameter values indexing its probability distributions. For an exponential distribution parametrized by its mean, $\Theta =(0,\infty )$; for a Bernoulli distribution, $\Theta =[0,1]$. Intersecting a confidence interval with the known parameter space cannot remove the true statistical parameter and therefore preserves its coverage probability.

Here the exponential distribution is parametrized by its mean, so $f(x;\theta )={\theta }^{-1}{e}^{-x/\theta }$ for $x\ge 0$ and $\theta >0$. Its log-likelihood and maximum-likelihood estimate areFor one observation, ${\partial }_{\theta }\log f=-1/\theta +X/{\theta }^2$. Since $\mathrm{Var}(X)={\theta }^2$, its Fisher information is $I_1(\theta )=1/{\theta }^2$. Consequently the estimated standard error is $\overline{X}/\sqrt{n}$, givingThis is a large-sample confidence interval, not an exact one. If its lower endpoint is negative, intersecting with the parameter space $(0,\infty )$ removes impossible values without excluding any possible true statistical parameter.