Check a Polynomial Density
Enter a polynomial and its support limits. The validator checks its exact area and the critical points that can produce its lowest value.
Example Data Table
| Function on the interval | Area | Nonnegative? | Result |
|---|---|---|---|
| f(x) = 0.5 on [0, 2] | 1 | Yes | Valid density |
| f(x) = 0.5x on [0, 2] | 1 | Yes | Valid density |
| f(x) = x on [0, 1] | 0.5 | Yes | Normalize by 2 |
| f(x) = x − 0.5 on [0, 1] | 0 | No | Invalid density |
Formula Used
A valid probability density must satisfy two conditions across its full support interval.
f(x) ≥ 0 for every x from L to U.
∫LU f(x) dx = 1
For f(x) = a3x3 + a2x2 + a1x + a0, the exact area is:
∫ f(x)dx = a3x4/4 + a2x3/3 + a1x2/2 + a0x
The page evaluates that antiderivative at U and L, then subtracts. It also tests endpoints and derivative roots inside the support. Those points locate the minimum of a cubic or lower polynomial on a closed interval.
When the function is nonnegative and has positive area A, its normalization factor is k = 1/A. The normalized density is g(x) = kf(x).
How to Use This Calculator
- Choose the highest power present in your polynomial.
- Enter the matching coefficients. Unused higher terms are ignored.
- Set the lower and upper support limits for the random variable.
- Add an evaluation point to inspect a local function value and cumulative result.
- Choose a small tolerance. The default handles common decimal rounding.
- Select Validate Density. Read the verdict above the form.
- Use the normalization factor only when the candidate stays nonnegative and has positive area.
- Download the CSV summary or use the print control for a PDF record.
This tool validates polynomial functions through degree three on a finite interval. It does not assess non-polynomial functions, infinite supports, or real-world model fit.
Understanding the Check
What a Probability Density Means
A probability density function describes a continuous random variable. It does not give a direct probability at one exact point. Instead, probability comes from an area under the curve. The function must stay nonnegative throughout its allowed interval. Its complete area must equal one. These two rules are essential. A curve can look reasonable yet fail either rule. This calculator checks both rules for a polynomial expression.
Why the Interval Matters
A density is always linked to a support interval. The same expression can be valid on one interval and invalid on another. For example, a linear expression may be positive between zero and two. It may become negative when the interval extends farther left. The selected lower and upper limits define every test. They also determine the total area. Use the true limits from your model. Do not select limits only because they produce a preferred result.
Checking Nonnegative Values
The first requirement is f(x) greater than or equal to zero. A polynomial can change direction within an interval. Checking only the endpoints is not always enough. This page evaluates endpoints and stationary points. Stationary points occur where the derivative equals zero. Those locations can reveal a hidden minimum. The smallest tested value is shown in the results. A value below the tolerance makes the proposed density invalid.
Checking Total Area
The second requirement is total area equal to one. The calculator integrates the polynomial exactly. It does not estimate the area from a graph. For a cubic polynomial, each term receives an antiderivative. The difference between the upper and lower evaluations gives the area. A positive area different from one does not automatically describe a valid density. However, it may be normalized. The result includes the multiplier needed to make its total area one.
Reading the Extra Measures
When the function is nonnegative and has positive area, the page also calculates useful distribution measures after normalization. The expected value estimates the balance point of the distribution. The second moment helps measure spread. Variance is the second moment minus the square of the mean. The evaluation point shows the function value and cumulative area up to a chosen position. These outputs help you inspect the shape beyond a simple valid or invalid verdict. It shows whether the entered point lies inside selected support, keeping cumulative probability interpretation consistent and avoiding extrapolation beyond the stated limits.
Using Results Carefully
A valid density is a mathematical condition, not proof that a model fits real data. Review the variable units and the interval meaning. Compare the distribution against observations when data is available. Use a sensible tolerance for rounded coefficients. Very small negative values may come from rounding. The calculator cannot replace judgment about discontinuities, excluded points, or changing support, so document assumptions before applying results to decisions, reports, or technical designs safely. Large negative values indicate a real problem. Change one coefficient at a time when exploring alternatives. Reliable density checks support clearer probability decisions every day.
Frequently Asked Questions
What makes a function a probability density?
It must never be negative across its support, and its total area over that support must equal one. Both conditions are required.
Can a density value be greater than one?
Yes. A density value can exceed one. Only area represents probability, so the full area across the support must still equal one.
Why is my function called normalizable?
It stays nonnegative and has positive area, but that area differs from one. Multiplying by the displayed factor creates a properly normalized density.
Why does the calculator test stationary points?
A polynomial may dip below zero between the endpoints. Stationary points can contain the lowest value, so testing them helps verify nonnegativity.
Does the evaluation point need to be inside the interval?
No. The calculator accepts any finite point. Values outside the support receive density zero, while cumulative probability becomes zero or one.
What tolerance should I use?
The default is suitable for many rounded decimal inputs. Use a smaller tolerance for precise coefficients and a slightly larger one when expected rounding is documented.
Can I test a constant density?
Yes. Select degree zero and enter the constant value. A uniform density on an interval has height one divided by interval width.
Does this work for infinite intervals?
No. This page evaluates finite support limits only. Infinite intervals require different integration methods and are outside this calculator’s scope.
Can I test a function with square roots or exponentials?
No. The validator is designed for polynomial functions through degree three. Use an appropriate symbolic or numerical tool for other function families.
What does variance show?
Variance measures spread around the mean after normalization. A larger variance usually indicates values are distributed farther from the expected value.
Is a valid density always a good model?
No. Validity only proves the mathematical density rules. You should still compare the model with observed data and domain knowledge.
Reliable density checks support clearer probability decisions every day.