Continuous Expected Value and Variance Solver

Math Probability • Continuous Probability Distributions

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

Compute E[X] and Var(X) for a continuous random variable from a PDF \(f(x)\) on \([a,b]\) using numerical integration.

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

Allowed: x, + − * / ( ) ^, commas, and functions: sin, cos, tan, exp, log, sqrt, abs, pow, min, max. Constants: pi, e. (Also accepts π, ×, ÷, and the minus sign −.)

Lower bound a Upper bound b Normalization Integration subintervals (even, Simpson)

Larger n → better accuracy but slower. Simpson’s rule uses an even number of subintervals.

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PDF on [a,b] (shaded integral area)

Animation ready

The animation sweeps the shaded area from a to b. The badge shows computed E[X] and Var(X).

Enter a PDF and bounds, then click “Calculate”.

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Continuous expected value and variance

Core definitions

For a continuous random variable \(X\) with PDF \(f(x)\) on an interval \([a,b]\), the expected value (mean) and variance can be computed from integrals:

\[ \begin{aligned} E[X] &= \int_a^b x\,f(x)\,dx \\ E[X^2] &= \int_a^b x^2\,f(x)\,dx \\ \mathrm{Var}(X) &= E[X^2] - (E[X])^2 \end{aligned} \]

A valid PDF must satisfy \(f(x)\ge 0\) on \([a,b]\) and \(\int_a^b f(x)\,dx = 1\). If your function is not normalized, you can normalize it by dividing by its total area:

\[ f_{\text{norm}}(x)=\frac{f(x)}{\int_a^b f(t)\,dt}. \]

Website tip: the graph shades the PDF area on \([a,b]\), which visually represents \(\int_a^b f(x)\,dx\).

Worked example: Uniform on [0,1]

Let \(X\sim \text{Uniform}(0,1)\), so \(f(x)=1\) for \(0\le x\le 1\).

Step 1. Compute \(E[X]\).

\[ \begin{aligned} E[X] &= \int_0^1 x\cdot 1\,dx \\ &= \left[\frac{x^2}{2}\right]_0^1 \\ &= \frac{1}{2}=0.5 \end{aligned} \]

Step 2. Compute \(E[X^2]\).

\[ \begin{aligned} E[X^2] &= \int_0^1 x^2\cdot 1\,dx \\ &= \left[\frac{x^3}{3}\right]_0^1 \\ &= \frac{1}{3} \end{aligned} \]

Step 3. Compute variance.

\[ \begin{aligned} \mathrm{Var}(X) &= E[X^2]-(E[X])^2 \\ &= \frac{1}{3}-\left(\frac{1}{2}\right)^2 \\ &= \frac{1}{12}\approx 0.0833 \end{aligned} \]

How the calculator integrates (numerical method)

When no closed-form integral is easy, the calculator uses numerical integration (composite Simpson’s rule). It approximates the area by fitting parabolas across small subintervals.

\[ \int_a^b g(x)\,dx \approx \frac{h}{3}\Big(g(x_0)+g(x_n)+4\sum_{\text{odd }i} g(x_i)+2\sum_{\text{even }i} g(x_i)\Big), \quad h=\frac{b-a}{n}. \]

University extension: higher moments \(E[X^k]=\int x^k f(x)\,dx\) and moment generating functions \(M_X(t)=E[e^{tX}]\) summarize distributions.

PDF on [a,b] (shaded integral area)

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