Calculator
Example Data Table
These examples show typical inputs and the type of probability you can compute.
| Distribution | Parameters | Mode | Bounds / x | Interpretation |
|---|---|---|---|---|
| Normal | μ=0, σ=1 | Between | a=-1, b=1 | Probability of being within one unit of the mean. |
| Exponential | λ=0.5 | Left tail | x=3 | Probability the waiting time is at most 3. |
| Uniform | a=10, b=20 | Right tail | x=18 | Probability a random value is at least 18. |
| Gamma | k=3, θ=2 | Between | a=0, b=8 | Probability a sum-like variable lies in [0,8]. |
| Beta | α=2, β=5 | Left tail | x=0.4 | Probability a proportion is at most 0.4. |
Formula Used
For a probability density function f(x), the area (probability) over an interval is:
Numeric Integration
This calculator approximates the integral using either the trapezoidal rule or Simpson’s rule. Simpson’s rule uses an even number of steps n and combines parabolic fits: ∫ ≈ (h/3)[f(x₀)+4f(x₁)+2f(x₂)+…+4f(xₙ₋₁)+f(xₙ)], where h=(b−a)/n.
How to Use This Calculator
- Select a distribution that matches your density curve.
- Choose the probability mode: Between, Left tail, or Right tail.
- Enter bounds (a, b) or a single x value for tails.
- Fill in the parameters required for your distribution.
- Pick an integration method and steps for accuracy.
- Press Submit to see results above the form.
- Download your results as CSV or PDF if needed.
Probability Area and Interpretation
For continuous variables, probability is measured by area, not by point values. This calculator integrates the density over your chosen interval and returns a value between 0 and 1. When you compute P(a ≤ X ≤ b), the result is the fraction of total mass inside that slice. With a standard normal distribution, mass concentrates near the mean and thins in the tails.
Distribution Options and Practical Use
Normal models measurement noise, Exponential models waiting times, Uniform represents bounded randomness, Gamma captures accumulated waiting, and Beta models proportions on [0,1]. Each option exposes only required parameters, reducing input mistakes. Common classroom settings use μ=0, σ=1; service systems often use λ from 0.1 to 2; and Beta(2,5) is a useful skewed proportion model.
Numerical Methods and Accuracy Controls
The tool applies numerical integration. Simpson’s rule is recommended because it typically converges faster for smooth densities than the trapezoidal rule. Steps n control resolution: larger n reduces discretization error but increases computation cost. For many intervals, n=1,000 is solid; narrow peaks or extreme parameters may benefit from 5,000 to 20,000 steps.
Handling Supports and Tail Limits
Bounded densities like Uniform[a,b] and Beta on [0,1] require bounds inside the support, so the calculator validates them. Unbounded curves need practical truncation for tail probabilities. For a normal curve, integrating over μ±10σ captures essentially all mass for reporting. For Exponential and Gamma, the upper limit extends using a conservative multiple of a scale estimate.
Results Table and Export Workflow
The results block summarizes the query, method, steps, and both requested and used bounds. A raw area is shown with a clamped value for quick sanity checks. CSV export supports spreadsheet logging, while PDF export creates a printable snapshot for assignments, lab notes, or documentation. Consistent fields make scenario comparisons faster.
Visualization for Verification
A Plotly graph helps you confirm the computed area visually. The density curve is drawn across a reasonable x-range, and the integrated region is shaded between the used bounds. If the shaded region looks wrong, adjust bounds, parameters, or steps. Visual checks often catch swapped bounds or unrealistic parameter scales early, reliably.
FAQs
Is the returned value always between 0 and 1?
Yes. The calculator reports a raw integral and a clamped probability. If inputs are valid, the clamped value stays within 0–1, matching probability rules.
Why can tail probabilities mention truncation?
Some distributions extend to infinity. The tool approximates infinite limits using safe finite bounds that capture nearly all mass for practical reporting, then integrates over that range.
Which integration method should I choose?
Simpson’s rule is usually best for smooth curves because it converges quickly as steps increase. Use trapezoids for quick rough checks or when experimenting with very large ranges.
How many steps are enough?
Start with 1,000 steps. Increase to 5,000–20,000 for narrow peaks, steep parameters, or when the shaded region is tiny. More steps can improve accuracy but may slow calculation.
What happens if my bounds are outside the support?
For bounded supports like Uniform and Beta, the calculator flags invalid bounds. For supported regions partly outside, use valid limits or change mode. This prevents misleading probabilities.
How does the graph relate to the computed area?
The curve shows f(x) and the shaded region matches the integration bounds used in computation. If the shading doesn’t match your intent, revise inputs and resubmit to recompute the area.