Continuous Pdf Validator Calculator

Math Probability • Continuous Probability Distributions

Written by STEM Calculators Team Published February 14, 2026 Updated February 24, 2026

Validate whether a proposed continuous PDF is legitimate on \([a,b]\): checks non-negativity and whether \(\int_a^b f(x)\,dx \approx 1\). Optionally normalize it, and compute \(E[X]\), \(Var(X)\).

Preset PDF (optional) PDF function \(f(x)\)

Allowed: \(x\), numbers, + − * / ^ ( ), commas, and functions: sin, cos, tan, exp, log, sqrt, abs, pow, min, max. Constants: pi, e. Use explicit multiplication like 2*x (not 2x). Multi-arg min/max supported, e.g. max(x, 0, 1).

Lower bound \(a\) Upper bound \(b\) Normalization Simpson subintervals (even)

Larger \(n\) → higher accuracy but slower. Simpson’s rule requires an even number (the tool will auto-fix to even).

Non-negativity test points

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PDF plot (area and validity)

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The badge inside the plot box shows the computed area \(\int_a^b f(x)\,dx\) (or \(\int_a^b g(x)\,dx\) if normalized). The vertical bounds show \(a\) and \(b\) with their numeric values.

Enter \(f(x)\), bounds \([a,b]\), then click “Calculate”.

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Theory

What is a continuous PDF?

A probability density function (PDF) \(f(x)\) describes the distribution of a continuous random variable \(X\). Probabilities come from areas under the curve, not point values (since \(P(X=x)=0\) for continuous \(X\)).

\[ P(c\le X\le d)=\int_c^d f(x)\,dx. \]

This calculator validates whether a proposed function can be a PDF on a finite interval \([a,b]\), and can optionally normalize it.

Two required PDF conditions on \([a,b]\)

For \(f(x)\) to be a valid PDF on \([a,b]\), it must satisfy:

\[ \textbf{(1) Non-negativity: } f(x)\ge 0 \text{ for all } x\in[a,b], \qquad \textbf{(2) Total area: } \int_a^b f(x)\,dx = 1. \]

The tool tests non-negativity by evaluating \(f(x)\) on a dense grid of points and reporting the minimum value found. It checks the area using numerical integration.

Important: if \(f(x)\) dips below 0 even on a small region, it is not a valid PDF.

Normalization: turning “almost a PDF” into a PDF

Sometimes a function is nonnegative but its integral is not 1. If

\[ I=\int_a^b f(x)\,dx \ne 1\quad \text{and}\quad I>0, \]

you can normalize it:

\[ g(x)=\frac{f(x)}{I}. \]

Then \(g(x)\ge 0\) (if \(f\ge 0\)) and:

\[ \int_a^b g(x)\,dx=\int_a^b \frac{f(x)}{I}\,dx=\frac{1}{I}\int_a^b f(x)\,dx=\frac{I}{I}=1. \]

This tool offers: Auto-normalize (only if the area differs from 1 beyond a tolerance) and Force normalization (always use \(g\)). If \(\int_a^b f(x)\,dx \le 0\), normalization is not possible.

Numerical integration (Simpson’s rule)

Many PDFs do not integrate neatly by hand, so the calculator uses Simpson’s rule on \([a,b]\). Split the interval into \(n\) equal subintervals (where \(n\) is even), with step size \(h=(b-a)/n\).

\[ \int_a^b f(x)\,dx \approx \frac{h}{3}\Big(f(x_0)+f(x_n)+4\!\!\sum_{\text{odd }i}\! f(x_i)+2\!\!\sum_{\text{even }i}\! f(x_i)\Big). \]

Simpson’s rule is accurate for smooth functions and converges quickly as \(n\) increases. If your function has sharp spikes or steep changes, increase \(n\) for better accuracy.

Mean and variance from a PDF

If \(f\) is a valid PDF (or if you normalize it to \(g\)), you can compute expectations by integration:

\[ E[X]=\int_a^b x\,f(x)\,dx,\qquad E[X^2]=\int_a^b x^2\,f(x)\,dx,\qquad Var(X)=E[X^2]-E[X]^2. \]

The calculator reports these values using the same numerical integration method, so they remain consistent with the area check.

Graph meaning: why the shaded area equals probability

On the graph, the shaded region under the curve represents an integral. Over the full interval \([a,b]\), that area is:

\[ \text{Area}=\int_a^b f(x)\,dx. \]

A valid PDF must have total shaded area equal to 1. If you only shade a sub-interval \([c,d]\subseteq[a,b]\), then the shaded area becomes the probability \(P(c\le X\le d)\).

University note: improper integrals and infinite intervals

Some PDFs are defined on infinite intervals, such as \([0,\infty)\) or \((-\infty,\infty)\). In those cases, the “area equals 1” condition becomes an improper integral, e.g.:

\[ \int_0^\infty f(x)\,dx = \lim_{t\to\infty}\int_0^t f(x)\,dx = 1. \]

This calculator focuses on finite \([a,b]\). For an improper-integral PDF, you can approximate by choosing a large cutoff interval (large \(b\)) and checking whether the area is close to 1, but this is only an approximation.

PDF plot (area and validity)

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