Analyze continuous distributions using tabulated x and densities. See moments, normalization, variance, and weighted plots. Download clean reports for classes, audits, and probability work.
This sample approximates a symmetric triangular density on the interval [0, 2]. The exact expected value is 1.
| x | f(x) | x·f(x) |
|---|---|---|
| 0.0 | 0.0 | 0.0 |
| 0.5 | 0.5 | 0.25 |
| 1.0 | 1.0 | 1.0 |
| 1.5 | 0.5 | 0.75 |
| 2.0 | 0.0 | 0.0 |
Continuous expectation: E[X] = ∫ x f(x) dx over the full support.
Normalization check: ∫ f(x) dx should equal 1 for a valid density.
Second moment: E[X²] = ∫ x² f(x) dx.
Variance: Var(X) = E[X²] − (E[X])².
This calculator uses the trapezoidal rule across the tabulated points. It approximates the areas under f(x), x·f(x), and x²·f(x).
It estimates the expected value of a continuous random variable from tabulated density values. It also reports total area, second moment, variance, standard deviation, and an approximate quantile from the same data.
A probability density cannot be negative. Negative values would create invalid probability mass and distort the area, expected value, and all higher moments.
Yes, a valid density integrates to one. This tool still computes E[X] by dividing weighted area by total area, then reports the normalization gap so you can see any mismatch quickly.
The tool applies the trapezoidal rule to the curve x·f(x). That numerical integral is divided by the integral of f(x), which gives an expected value estimate from your tabulated points.
Yes. The x values do not need equal spacing. They only need to be strictly increasing so each interval width remains positive for the integration steps.
It is the approximate x-value where the cumulative probability reaches your chosen probability level. For example, 0.50 returns an estimated median from the tabulated density.
The graph helps you inspect the density shape, the weighted curve x·f(x), and the cumulative distribution. Visual checks often reveal skewness, clipping, sparse sampling, or normalization issues.
Use more points when the density changes rapidly or has curved sections. A denser grid usually improves the trapezoidal approximation and reduces numerical error.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.