Calculator Inputs
Use the responsive grid below. Large screens show three columns, medium screens show two, and mobile screens show one.
Example Data Table
| Distribution | Inputs | Query | Illustrative Result |
|---|---|---|---|
| Standard Normal | x = 1.25 | P(X ≤ 1.25) | About 0.8944 |
| Normal | μ = 50, σ = 8, x = 62 | P(X ≤ 62) | About 0.9332 |
| Exponential | λ = 2.5, x = 0.8 | P(X ≤ 0.8) | About 0.8647 |
| Uniform | a = 10, b = 30, x = 22 | P(X ≤ 22) | 0.6000 |
| Binomial | n = 12, p = 0.40, k = 5 | P(X ≤ 5) | About 0.6652 |
| Poisson | λ = 4.5, k = 6 | P(X ≤ 6) | About 0.8311 |
Formula Used
General CDF Definition
F(x) = P(X ≤ x). For interval probabilities, continuous models use F(upper) − F(lower). Discrete models use F(upper) − F(lower − 1).
Standard Normal and Normal
F(x) = 0.5 × [1 + erf((x − μ) / (σ√2))]. The standard normal model fixes μ = 0 and σ = 1.
Exponential
F(x) = 1 − e−λx for x ≥ 0. When x < 0, the cumulative probability is zero.
Uniform
F(x) = 0 for x < a, F(x) = (x − a)/(b − a) for a ≤ x ≤ b, and F(x) = 1 for x > b.
Binomial
F(k) = Σ C(n,i)pi(1−p)n−i from i = 0 to k.
Poisson
F(k) = Σ e−λ λi / i! from i = 0 to k.
How to Use This Calculator
- Choose the distribution that matches your problem.
- Select a left tail, right tail, or interval query.
- Enter the required parameter values such as mean, rate, limits, or trial settings.
- Provide the target x value, or use lower and upper bounds for interval probability.
- Click Calculate CDF to show the result directly under the header and above the form.
- Review the summary cards, detailed result table, and graph.
- Export the output with CSV or PDF buttons.
Frequently Asked Questions
1. What does a cumulative distribution function measure?
It gives the probability that a random variable is less than or equal to a chosen value. It accumulates probability from the left side of the distribution up to that point.
2. When should I use a standard normal model?
Use it when values are already standardized into z-scores, or when your variable naturally has mean zero and standard deviation one.
3. What is the difference between left-tail and right-tail probability?
Left-tail probability measures values at or below the chosen point. Right-tail probability measures values at or above the chosen point.
4. Why does the interval formula differ for discrete distributions?
Discrete models count isolated outcomes, so interval probability must remove all values below the lower integer. That is why the calculator uses F(upper) − F(lower − 1).
5. What does the complement value represent?
It is one minus the reported event probability. This helps when you want the opposite event, such as values outside a selected interval.
6. Why do some inputs disappear after changing the distribution?
Each distribution needs different parameters. The interface hides irrelevant fields so the form stays cleaner and easier to complete.
7. Is the graph showing density or cumulative probability?
The graph shows cumulative probability. It rises from zero toward one as x or k increases across the distribution support.
8. Can I use this calculator for teaching and reports?
Yes. The result table, chart, and export buttons make it useful for homework checks, audit notes, working papers, and classroom explanations.