Enter the observed number of events to calculate the 90%, 95%, and 99% confidence intervals for a Poisson-distributed event rate.

## Understanding Poisson Confidence Intervals

This calculator provides confidence intervals for the rate of a Poisson-distributed event, estimating the range within which the true rate is likely to fall. Given an observed count of events, different methods are available to calculate the confidence interval at specific confidence levels.

## Calculation Methods

The confidence interval for a Poisson mean \( k \) (observed events) can be estimated using various methods. Here are the primary approaches:

### 1. Chi-Squared Approximation

This method uses the Chi-Squared distribution to approximate the interval, providing accurate bounds for Poisson means. The formula for the interval is:

Lower Bound: \( \frac{1}{2} \chi^2_{\alpha/2, 2k} \)

Upper Bound: \( \frac{1}{2} \chi^2_{1 - \alpha/2, 2k + 2} \)

where \( \chi^2_{p, \nu} \) represents the Chi-Squared quantile function with probability \( p \) and degrees of freedom \( \nu \), and \( \alpha \) is the confidence level. This method provides an accurate and commonly used approximation, especially for larger values of \( k \).

### 2. Normal Approximation

For large values of \( k \), the Poisson distribution resembles a normal distribution. Using this approximation, we can calculate the confidence interval as:

\( CI = k \pm Z \cdot \sqrt{k} \)

where \( Z \) is the Z-score corresponding to the confidence level (e.g., 1.645 for 90%, 1.96 for 95%). While this approximation is simpler, it becomes less accurate for small values of \( k \) where the Poisson distribution is more skewed.

### 3. Exact Method Using Poisson Quantiles

The exact method provides a confidence interval by directly using the quantiles of the Poisson distribution, without any approximation. In this approach:

- The
**Lower Bound**is the smallest integer \( L \) such that the cumulative probability \( P(X \leq L) \geq \alpha/2 \) (for a two-tailed test). - The
**Upper Bound**is the largest integer \( U \) such that the cumulative probability \( P(X \leq U) \leq 1 - \alpha/2 \).

For example, for a 95% confidence interval with observed events \( k = 20 \), we find the values of \( L \) and \( U \) that meet the required cumulative probabilities in the Poisson distribution. This method provides the most accurate interval, but it requires calculating cumulative Poisson probabilities, which can be computationally intensive.

Suf is a senior advisor in data science with deep expertise in Natural Language Processing, Complex Networks, and Anomaly Detection. Formerly a postdoctoral research fellow, he applied advanced physics techniques to tackle real-world, data-heavy industry challenges. Before that, he was a particle physicist at the ATLAS Experiment of the Large Hadron Collider. Now, he’s focused on bringing more fun and curiosity to the world of science and research online.