Distribution Curve Calculator

Calculator inputs

Choose a model, enter its parameters, then evaluate a point and interval. Results appear above this form after submission.

Distribution type

Switch the curve type here.

Mean μ

Center of the normal curve.

Standard deviation σ

Spread of the normal curve.

Trials n

Number of fixed Bernoulli trials.

Success probability p

Chance of success on one trial.

Rate λ

Event rate or decay rate.

Point x or k

Evaluate density or probability at this value.

Lower bound A

Used for interval probability calculations.

Upper bound B

The calculator swaps A and B if needed.

Example data table

Distribution	Inputs	Point result	Cumulative result	Interval result
Normal	μ = 50, σ = 10, x = 65, A = 40, B = 60	f(65) = 0.012952	F(65) = 0.933193	P(40 ≤ X ≤ 60) = 0.682689
Binomial	n = 20, p = 0.40, k = 8, A = 6, B = 10	P(X = 8) = 0.179706	P(X ≤ 8) = 0.595599	P(6 ≤ X ≤ 10) = 0.746880
Poisson	λ = 4.5, k = 3, A = 2, B = 6	P(X = 3) = 0.168718	P(X ≤ 3) = 0.342296	P(2 ≤ X ≤ 6) = 0.769951
Exponential	λ = 0.8, x = 2, A = 1, B = 3	f(2) = 0.161517	F(2) = 0.798103	P(1 ≤ X ≤ 3) = 0.358609

Formula used

Normal distribution

Density: f(x) = [1 / (σ√(2π))] × e^{-0.5((x-μ)/σ)²}

Cumulative: F(x) is estimated from the error function approximation. Mean = μ. Variance = σ².

Binomial distribution

Probability mass: P(X = k) = C(n,k) × p^k × (1-p)^n-k

Moments: Expected value = np. Variance = np(1-p). The cumulative value sums masses from 0 through k.

Poisson distribution

Probability mass: P(X = k) = e^-λ × λ^k / k!

Moments: Expected value = λ. Variance = λ. The cumulative value sums masses from 0 through k.

Exponential distribution

Density: f(x) = λe^-λx, for x ≥ 0

Cumulative: F(x) = 1 - e^-λx. Expected value = 1/λ. Variance = 1/λ².

How to use this calculator

Select the distribution that matches your problem structure.
Enter the required parameters, such as mean, spread, trials, or rate.
Enter the evaluation point in the x or k field.
Provide A and B to measure the probability across an interval.
Submit the form and review the result cards above the form.
Use the chart to see shape, concentration, and interval emphasis.
Download a CSV or PDF summary if you need a record.

FAQs

1. What does the density value mean?

Density describes curve height at one point. For continuous models, it is not the same as exact point probability. Use interval probability for a meaningful chance between two values.

2. When should I use the normal distribution?

Use it for symmetric data centered around a mean, especially when many small effects combine. Measurements, scores, and process variation often fit it reasonably well.

3. What is the difference between PMF and CDF?

PMF gives probability at a discrete count. CDF gives cumulative probability up to a value. Continuous models use density instead of PMF for single-point evaluation.

4. Why are interval bounds rounded for discrete models?

Binomial and Poisson outcomes occur as whole counts. The calculator rounds interval bounds to integers so the reported probability matches the actual support.

5. Can I use large binomial trial counts?

Yes, but very large trial counts can slow exact summation. This page limits trials to practical ranges to keep calculations responsive and stable.

6. What does right-tail probability show?

It reports the chance of observing a value at least as extreme on the high side. This is useful for thresholds, alarms, and exceedance studies.

7. Why must spread and rate values stay positive?

These parameters control spread or decay. Zero or negative values break the model definitions and produce invalid densities, cumulative values, and chart shapes.

8. Is this calculator suitable for final research reporting?

It is strong for teaching, screening, and quick checks. For formal research, verify assumptions, rounding rules, and numerical precision with your statistical workflow.