a and b. For quantiles, enter p and use inverse CDF.
| Distribution | Inputs | Operation | Result (approx.) |
|---|---|---|---|
| Normal | μ=50, σ=10, a=40, b=60 | P(a<X<b) | 0.682689 |
| Uniform | a=0, b=10, x=3 | CDF at x | 0.300000 |
| Exponential | λ=0.5, x=2 | PDF at x | 0.183940 |
| Standard Normal | p=0.975 | Inverse CDF | 1.959964 |
f(x)= (1/(σ√(2π))) · e^{-((x-μ)²)/(2σ²)}CDF uses an error-function approximation.
f(x)=1/(b-a) for a≤x≤bF(x)=(x-a)/(b-a) inside the range.
f(x)=λe^{-λx} for x≥0F(x)=1-e^{-λx} for x≥0.
F(b) − F(a). Inverse CDF returns the value x such that F(x)=p.
- Select a distribution that matches your data model.
- Choose an operation: PDF, CDF, interval probability, or quantile.
- Enter only the inputs required for that operation.
- Provide the distribution parameters in the appropriate fields.
- Press Calculate to view results above the form.
- Use the download buttons to export the latest summary.
Density Curves Guide
1) What this density curves tool does
Density curves describe how values are distributed across a range. This calculator evaluates standard curves and outputs point density f(x), cumulative probability P(X ≤ x), and interval probability P(L ≤ X ≤ U) as areas under the curve. You can also request left-tail, right-tail, or between-bounds probabilities, with input validation for missing limits and range errors quickly safely too.
2) The key rule: total area equals 1
A valid density curve is never negative and has total area 1. Any area over an interval becomes a probability. An area of 0.25 between two x-values means a 25% chance that X lies there.
3) Curve options
Normal: symmetric, controlled by mean μ and standard deviation σ. Uniform: constant likelihood between minimum a and maximum b. Exponential: waiting-time model with rate λ, defined for x ≥ 0 and declining as x increases.
4) Inputs that change shape
Normal: raising μ shifts the center; raising σ spreads values and lowers the peak. Uniform: widening (b − a) lowers height 1/(b − a). Exponential: higher λ concentrates probability near zero and shortens typical waits.
5) Formulas used
Normal: f(x)= (1/(σ√(2π)))·exp(−(x−μ)²/(2σ²)). Uniform: f(x)=1/(b−a) for a≤x≤b, else 0. Exponential: f(x)=λ·exp(−λx) for x≥0, else 0. Probabilities come from integrating f(x) over your limits.
6) How to interpret results
Probability is area, not height. A tall curve can still give a small probability over a narrow interval. For normal curves, the z-score z=(x−μ)/σ shows how many standard deviations x is from μ.
7) Quick data checks
Use the example table to test sensitivity. If μ=50, σ=10, and x=60, then z=1. If σ becomes 20, z drops to 0.5 and the peak flattens. Uniform: changing a=0, b=100 to a=0, b=200 halves the height. Exponential: mean wait is 1/λ, so λ=0.2 gives 5, while λ=0.5 gives 2. These checks help you choose realistic parameters before analysis.
FAQs
1) What does f(x) represent?
f(x) is the curve height at x. It shows relative concentration near x, but it is not a probability by itself unless multiplied by a very small width.
2) Why can a probability be zero while f(x) is positive?
For continuous curves, P(X = x) is zero because a single point has zero width. Probabilities come from areas over intervals.
3) When should I pick a normal curve?
Use it for measurements shaped by many small effects, like heights or errors. It works best when the data look roughly symmetric around a center.
4) What limits apply to the uniform curve?
Uniform probabilities are only valid between a and b. Outside that range, density is zero and the calculator will return zero area.
5) How do I interpret the exponential rate λ?
λ controls how fast the curve declines. The mean waiting time is 1/λ, so larger λ means shorter expected waits.
6) Why do my results change when I widen the interval?
Wider intervals capture more area under the curve, so probability increases. If you narrow the interval, you capture less area even if the curve is tall.