Calculator
Example Data Table
| Case | Function | Support | x | Expected CDF idea |
|---|---|---|---|---|
| Uniform shape | 0.5 | [0, 2] | 1 | Area grows linearly. |
| Triangular shape | x | [0, 1] | 0.7 | Normalize because total area is 0.5. |
| Exponential shape | exp(-x) | [0, 8] | 2 | Most mass is near zero. |
| Bell shaped sample | exp(-0.5*x^2) | [-5, 5] | 1.96 | Normalization converts shape into density. |
Formula Used
Normalized density: g(x) = f(x) / A, where A = integral from a to b of f(t) dt.
Cumulative value: F(x) = integral from a to x of g(t) dt.
Interval probability: P(c ≤ X ≤ d) = F(d) - F(c).
Survival value: S(x) = 1 - F(x), when the density is normalized.
Mean: E[X] = integral from a to b of x g(x) dx.
Variance: Var(X) = integral from a to b of (x - mean)^2 g(x) dx.
How to Use This Calculator
Enter a function using x as the variable. Add multiplication signs where needed, such as 2*x.
Set finite lower and upper support values. These bounds define the area used for the CDF.
Enter the target x value. Add an interval if you need probability between two values.
Keep normalization enabled when your function is only a density shape. Select Simpson for smoother functions.
Press the calculate button. The result appears below the header and above the form.
CDF From Function Calculator Guide
What This Calculator Does
A cumulative distribution function shows how much probability has gathered up to a selected value. This tool starts with a user supplied density function. It then integrates that function across a finite support. The result is a CDF value, an interval probability, and useful summary measures.
Why CDF Values Matter
CDF values help compare positions inside a distribution. A value near zero means the point is close to the lower tail. A value near one means the point is close to the upper tail. Engineers, analysts, teachers, and students use this view when checking risk, reliability, waiting time, demand, and measurement spread.
Function Based Input
Many calculators only support named distributions. This one accepts a custom expression in x. You can enter a polynomial, exponential curve, trigonometric term, or a mixed formula. Use a finite lower bound and upper bound. The calculator checks the total area and can normalize it. This is helpful when your function has the right shape but not unit area.
Numerical Integration
The calculator uses trapezoid or Simpson integration. Trapezoid integration is simple and stable. Simpson integration is often more accurate for smooth curves. More steps can improve accuracy, but very large step counts may slow the page. The displayed total area helps confirm whether the density is valid over the selected support.
Advanced Outputs
The result includes the PDF value at x, CDF at x, survival value, interval probability, mean, variance, and standard deviation. A percentile option estimates the x value where the CDF reaches a target probability. This is useful for cutoffs, grades, quality limits, and service thresholds.
Practical Tips
Always check that the density stays nonnegative. A probability density should not drop below zero. Also confirm the support covers the full range you intend to study. If the total mass is far from one, keep normalization enabled. For sharp curves, use more steps. For functions with breaks, choose bounds around one smooth section, or test each section separately.
Learning Value
This calculator connects algebra, calculus, and probability. It shows how area under a curve becomes cumulative probability. It also makes assumptions visible. That makes it useful for homework, reports, modeling checks, and quick experiments with custom density functions online.
FAQs
1. What is a CDF?
A CDF gives the probability that a random variable is less than or equal to a selected value.
2. What function should I enter?
Enter a density function in x. It can be a complete density or a shape that needs normalization.
3. Why should I normalize the function?
Normalization scales the total area to one. A valid probability density must have total area equal to one.
4. Can I use infinite bounds?
This calculator uses finite numerical integration. Choose practical lower and upper bounds that capture the distribution mass.
5. Which method is more accurate?
Simpson rule is usually better for smooth curves. Trapezoid rule is simple and useful for rough checks.
6. What does survival value mean?
The survival value is the remaining probability above x. For normalized densities, it is one minus the CDF.
7. Why do I see a negative density warning?
A probability density should not be negative. Review the function, support, and typed operators carefully.
8. Can I download the results?
Yes. Use the CSV button for spreadsheet work. Use the PDF button for a simple saved report.