Calculator Inputs
Formula Used
Interval probability: P(a ≤ X ≤ b) = F(b) - F(a).
Quantile: Q(p) solves F(x) = p.
Normal: f(x) = exp(-z²/2) / (σ√2π), where z = (x - μ) / σ.
Uniform: f(x) = 1 / (b - a) for values inside the support.
Exponential: f(x) = λ exp(-λx), and F(x) = 1 - exp(-λx).
Gamma: f(x) = xk-1 exp(-x/θ) / (Γ(k)θk).
Beta: f(y) = yα-1(1-y)β-1 / B(α,β), scaled to the support range.
Lognormal: f(x) = exp(-(ln(x)-μ)² / (2σ²)) / (xσ√2π).
How to Use This Calculator
- Select the continuous distribution that matches your problem.
- Enter the value x for density and cumulative probability.
- Enter lower and upper bounds for interval probability.
- Enter p to find the matching quantile value.
- Fill only the parameter fields for the selected distribution.
- Press Calculate to view results above the form.
- Use the export buttons to save results.
Example Data Table
| Distribution | Parameters | x | Interval | p | Use Case |
|---|---|---|---|---|---|
| Normal | mean 0, sd 1 | 1 | 0 to 1.96 | 0.95 | Standard score analysis |
| Uniform | min 0, max 10 | 4 | 2 to 7 | 0.8 | Equal chance range |
| Exponential | rate 0.5 | 3 | 1 to 5 | 0.9 | Waiting time model |
| Gamma | shape 2, scale 2 | 5 | 2 to 8 | 0.75 | Flexible positive data |
Continuous Probability Distribution Guide
Why Continuous Models Matter
Continuous probability models describe variables that can take any value inside a range. Common examples include height, delivery time, rainfall, lifespan, error size, and machine output. A single point has almost zero probability. The useful value is area under the curve. That area gives the chance that a future observation falls inside a selected interval.
Supported Distribution Types
This calculator compares six important families. The normal model suits balanced variation around a mean. The uniform model gives equal density across a fixed span. The exponential model studies waiting time before an event. The gamma model extends waiting time into flexible shapes. The beta model fits percentages inside a bounded range. The lognormal model fits positive skewed measurements.
Reading the Output
Each result is built from density, cumulative probability, interval area, quantile, and moment formulas. Density shows how concentrated the curve is near the selected value. The cumulative value shows the probability below that value. Interval probability subtracts two cumulative values. A quantile reverses that process. It finds the value linked with a chosen cumulative probability.
Choosing Better Inputs
Good inputs matter. Select a distribution that matches the data source. Use positive scale values. Keep beta support limits ordered. Use interval limits that reflect the real question. A delivery estimate may need a right tail. A quality tolerance may need a middle interval. A risk limit may need a left tail.
Using Results in Reports
The calculator also reports mean, variance, standard deviation, and median. These statistics make reports easier to compare. They help explain center, spread, and typical outcomes. Export buttons create files for records and sharing. Use the example table to test the workflow first. Then replace those values with project data.
Planning With Probabilities
For planning work, interval probability often gives the most useful answer. It can show the chance of finishing between two dates. It can estimate the percent of parts inside tolerance. It can measure the likelihood of demand staying below capacity. Quantiles are also practical. A ninety fifth percentile gives a conservative planning limit. A median gives a central target. Together, these outputs turn a curved model into simple figures that managers, students, analysts, and engineers can discuss without losing the statistical meaning. Always compare modeled results with observed data before using them for costly, safety, or policy choices in practice.
FAQs
What is a continuous probability distribution?
It models outcomes that can take any value across a range. Probability is measured as area under the curve, not at one exact point.
Which distribution should I select?
Choose normal for symmetric data, exponential for waiting times, gamma for flexible positive data, beta for bounded percentages, and uniform for equal ranges.
What does density f(x) mean?
Density shows curve height at x. It is not usually a direct probability. Interval area gives the actual probability.
What does cumulative F(x) mean?
It is the probability that a random value is less than or equal to x. It always ranges from zero to one.
How is interval probability calculated?
The calculator subtracts the lower cumulative value from the upper cumulative value. This gives the area between both limits.
What is a quantile?
A quantile is the value connected to a chosen cumulative probability. For example, p equals 0.95 gives the ninety fifth percentile.
Can I export the calculation?
Yes. Use the CSV button for spreadsheet work. Use the PDF button for a simple printable report.
Are the results exact?
Normal, gamma, beta, and lognormal functions use numerical approximations. They are suitable for practical calculations and routine statistical reporting.