From point estimate to interval estimate
A point estimate gives one number, such as $\bar X$. An interval estimate gives a plausible range for the unknown parameter and reports the uncertainty caused by sampling variation.
In a frequentist confidence interval, the confidence level describes the long-run success rate of the procedure. In a Bayesian credible interval, the probability statement is about the posterior distribution of the parameter.
Core formulas to compare
| Model | Parameter | Typical interval idea |
|---|---|---|
| Normal, known $\sigma$ | $\mu$ | $\bar X\pm z_{1-\alpha/2}\sigma/\sqrt n$ |
| Poisson | $\lambda$ | Invert exact count distribution |
| Exponential mean | $\mu$ | Use $2\sum X_i/\mu\sim\chi^2_{2n}$ |
| Bayesian Poisson | $\lambda$ | Gamma posterior quantiles |
Known $\sigma$ confidence interval
Use this when $X_1,\dots,X_n\sim N(\mu,\sigma^2)$ with known $\sigma$, or as a large-sample approximation.
Sampling distribution picture
The shaded central region has area equal to the confidence level under the standard normal curve.
What does “95% confidence” mean?
If we repeated the sampling process many times and built an interval by the same rule each time, about 95% of those intervals would contain the true parameter.
After the data are observed, the fixed frequentist interval either contains the true parameter or it does not. The 95% refers to the procedure, not to a posterior probability for this realized interval.
Bayesian credible intervals are different: a 95% credible interval can be read as containing 95% posterior probability for the parameter, conditional on the model, prior, and observed data.
Poisson mean $\lambda$
Let $X_1,\dots,X_n\sim\text{Poisson}(\lambda)$ and $S=\sum_i X_i$. Then $S\sim\text{Poisson}(n\lambda)$. A classical exact interval for $\lambda$ is obtained by inverting the count distribution.
The displayed chi-square quantiles use a fast Wilson-Hilferty approximation, which is excellent for teaching and visualization.
Poisson likelihood and interval
The curve is the likelihood shape as a function of $\lambda$; vertical lines mark the interval endpoints and MLE $\hat\lambda=S/n$.
Exact CI for mean $\mu$
Let $X_i\sim\mathrm{Exp}(\mu)$ where $\mu$ is the mean. Since $2\sum X_i/\mu\sim\chi^2_{2n}$, an exact confidence interval is
Pivot distribution
The central chi-square area determines the interval for $\mu$ by solving inequalities for $\mu$.
Poisson-Gamma posterior
For $X_i\sim\text{Poisson}(\lambda)$ with prior $\lambda\sim\text{Gamma}(a,b)$ using rate $b$, the posterior is
Posterior density and credible interval
The shaded posterior area is the equal-tail credible interval. Because $\text{Gamma}(k,r)=\chi^2_{2k}/(2r)$, the calculator uses chi-square quantiles.
Do confidence intervals cover?
Simulate many samples from a normal population and build the known-$\sigma$ interval each time.
Last 60 simulated intervals
Blue intervals cover the true mean. Red intervals miss it.
Percentile bootstrap for a mean
When a closed-form pivot is unavailable, the bootstrap approximates the sampling distribution by resampling from the observed data.
- Resample $n$ observations with replacement from the data.
- Compute the statistic for each bootstrap sample.
- Use empirical quantiles of bootstrap statistics as interval endpoints.
Bootstrap distribution of $\bar X^*$
The histogram shows bootstrap means; vertical lines show the percentile interval.