Probability Density Integrator

Math Calculus • Applications of Integrals

Written by STEM Calculators Team Published December 28, 2025 Updated February 24, 2026

9. Probability Density Integrator

Computes normalization \(\int_a^b f(x)\,dx\), expected value \(E[X]=\int_a^b x f(x)\,dx\) (with renormalization if needed), variance, and CDF values. Supports \(\pm\infty\) bounds via numeric transforms. \(\; \mu=E[X]=\dfrac{\int_a^b x f(x)\,dx}{\int_a^b f(x)\,dx},\quad \mathrm{Var}(X)=E[X^2]-\mu^2 \;\)

Inputs

PDF \(f(x)\)

Use variable x. Constants: pi, e, inf. Supported: + − * / ^, parentheses, sin cos tan ln log sqrt abs exp. Implicit multiplication allowed: 2x, (x+1)(x-1).

Lower bound \(a\)

Use -inf or inf if needed.

Upper bound \(b\)

Example: \(b=1\), \(b=\pi\), or \(b=\infty\).

Integration method

Adaptive is best for sharp peaks / tails.

Simpson segments \(N\)

Must be even. Used by Composite Simpson (and for CDF curve sampling).

Tolerance \(\varepsilon\)

Used only by Adaptive Simpson.

Normalize check tol

Warn if \(\big|\int f - 1\big|>\) tol.

CDF point \(x_0\)

Computes \(F(x_0)=\int_a^{x_0} f(x)\,dx\).

Options

Presets

Click a preset to load and evaluate.

Ready

Graph

Drag to pan • wheel/pinch to zoom • PDF is centered at \(x=0\) axis; CDF uses a right-side \(0\to 1\) scale.

\(f(x)\) shade \([a,b]\) CDF \(F(x)\) \(\mu\)

Center \(c\)

Graph window center (x-axis).

Half-width \(w\)

Graph spans \([c-w,c+w]\).

x: 0, y: 0, zoom(px/unit): 60

Result & Steps

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

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9. Probability Density Integrator — Theory

What this tool computes

A probability density function (PDF) \(f(x)\) must satisfy:

\[ f(x)\ge 0,\qquad \int_{a}^{b} f(x)\,dx = 1. \]

This calculator checks the normalization and then computes moments, mean, variance, and CDF values (renormalizing if \(\int f \ne 1\)).

Normalization

The normalization constant is:

\[ I_0 = \int_a^b f(x)\,dx. \]

If \(I_0\neq 1\), the tool reports a warning and uses the renormalized density \(\tilde f(x)=\dfrac{f(x)}{I_0}\) for \(E[X]\), \(\mathrm{Var}(X)\), and \(F(x)\).

Expected value and variance

The mean (expected value) is:

\[ \mu = E[X] = \int_a^b x\,\tilde f(x)\,dx = \frac{\int_a^b x f(x)\,dx}{\int_a^b f(x)\,dx}. \]

The second moment and variance are:

\[ E[X^2] = \int_a^b x^2\,\tilde f(x)\,dx = \frac{\int_a^b x^2 f(x)\,dx}{\int_a^b f(x)\,dx}, \qquad \mathrm{Var}(X)=E[X^2]-\mu^2. \]

Standard deviation is \(\sigma=\sqrt{\mathrm{Var}(X)}\).

CDF (cumulative distribution function)

The CDF is the area under the (normalized) density from the lower bound up to \(x\):

\[ F(x)=P(X\le x)=\int_a^x \tilde f(t)\,dt = \frac{\int_a^x f(t)\,dt}{\int_a^b f(t)\,dt}. \]

It always satisfies \(0\le F(x)\le 1\) and is non-decreasing.

Common presets and results

Distribution	PDF \(f(x)\)	Support	\(\mu\)	\(\mathrm{Var}(X)\)
Uniform	\(1\)	\([0,1]\)	\(\frac12\)	\(\frac{1}{12}\)
Exponential (\(\lambda=1\))	\(e^{-x}\)	\([0,\infty)\)	\(1\)	\(1\)
Standard normal	\(\frac{1}{\sqrt{2\pi}}e^{-x^2/2}\)	\((-\infty,\infty)\)	\(0\)	\(1\)

For custom PDFs, always check that \(\int_a^b f(x)\,dx\approx 1\) and that tails are integrable when bounds are infinite.

Frequently Asked Questions

What does a probability density integrator calculate?

It evaluates integrals of a PDF f(x) over [a,b], including the normalization I0 = ∫[a,b] f(x) dx, the mean μ, the variance, and the CDF value F(x0). These quantities describe total probability, average value, spread, and cumulative probability.

How is the expected value computed from a PDF?

The mean is computed as μ = (∫[a,b] x f(x) dx) / (∫[a,b] f(x) dx). If the PDF is not perfectly normalized, dividing by the normalization integral renormalizes the result.

Why does the normalization integral sometimes not equal 1?

For a valid PDF, the integral over its support should be 1, but numeric integration, sharp peaks, long tails, or an incorrect formula/bounds can produce I0 ≠ 1. The tool warns based on a tolerance and uses a renormalized density f(x)/I0 for moments and CDF values.

How does the calculator handle infinite bounds like -inf to inf?

It supports ±infinity bounds by applying numeric transforms that map infinite intervals to finite ones for integration. This is useful for distributions such as the normal distribution on (-inf, inf) or exponential tails on [0, inf).

When should I use Composite Simpson vs Adaptive Simpson?

Composite Simpson uses a fixed even number of segments N and works well for smooth functions over finite intervals. Adaptive Simpson refines intervals based on a tolerance ε and is often better for sharp peaks, discontinuities in derivatives, or heavy tails.

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