Calculator inputs
Choose a distribution, select a cumulative mode, and calculate a lower tail, upper tail, or interval probability.
CDF plot
The chart displays the cumulative distribution function for the chosen model. Submitted points or interval limits appear as highlighted markers.
Example data table
| Example | Distribution | Inputs | Query | Approximate result |
|---|---|---|---|---|
| Exam scores | Normal | μ = 70, σ = 10 | P(X ≤ 82) | 0.8849 |
| Defective parts | Binomial | n = 20, p = 0.08 | P(X ≤ 2) | 0.9216 |
| Calls per minute | Poisson | λ = 4 | P(2 ≤ X ≤ 6) | 0.7977 |
| Service life | Exponential | λ = 0.5 | P(X ≥ 3) | 0.2231 |
| Arrival time | Uniform | a = 0, b = 30 | P(5 ≤ X ≤ 18) | 0.4333 |
Formula used
Normal: Cumulative probability uses F(x) = 0.5 × [1 + erf((x − μ) / (σ√2))]. Interval probability equals F(b) − F(a).
Binomial: P(X ≤ x) = Σ C(n,k) pk (1−p)n−k, summed from k = 0 to x.
Poisson: P(X ≤ x) = Σ e−λ λk / k!, summed from k = 0 to x.
Exponential: F(x) = 1 − e−λx for x ≥ 0. The upper tail is e−λx.
Uniform: For support [a,b], F(x) = 0 below a, 1 above b, and (x−a)/(b−a) inside the interval.
How to use this calculator
- Select the probability distribution that best matches your data or process.
- Choose a cumulative mode for a lower tail, upper tail, or interval probability.
- Enter the required parameters, such as mean and standard deviation or trials and success rate.
- Provide the target x value or interval bounds.
- Set the number of decimal places, then submit the form.
- Review the probability, complement, odds, summary table, and chart.
- Export the current result as CSV or PDF when you need documentation.
FAQs
1. What does cumulative probability mean?
Cumulative probability adds all probability up to a point. For interval mode, it measures the probability between two bounds instead of only below one value.
2. When should I use the normal model?
Use the normal model for continuous data that is roughly symmetric and bell-shaped, such as test scores, measurement errors, or many natural variations.
3. Why are binomial and Poisson inputs treated as counts?
Both models describe discrete events. Their cumulative probabilities are built by summing whole-number outcomes, so decimal x values are converted to practical count limits.
4. What is the difference between lower and upper tail probability?
Lower tail probability measures outcomes at or below x. Upper tail probability measures outcomes at or above x, which is useful for exceedance and risk questions.
5. Why does a continuous exact point probability not appear here?
For continuous distributions, the probability at one exact point is zero. That is why cumulative and interval probabilities are more meaningful for normal, exponential, and uniform models.
6. What does the complement value show?
The complement is 1 minus the reported probability. It helps you interpret the opposite event immediately without repeating the full calculation.
7. Can I use this calculator for teaching or reports?
Yes. The result summary, example table, formula notes, chart, and export buttons make it useful for class demonstrations, technical notes, and audit trails.
8. How accurate are the results?
The results are appropriate for practical analysis and education. Normal values rely on a standard error-function approximation, while discrete models are summed directly.