CalcRef

Applications of Integrals · Centroids & Averages

Probability (Continuous)

P(a≤X≤b)=∫abf(x) dxwhere ∫−∞∞f(x) dx=1P(a \leq X \leq b) = \int_a^b f(x)\, dx \quad \text{where } \int_{-\infty}^{\infty} f(x)\, dx = 1

For a continuous random variable with probability density function f(x), the probability that X falls between a and b is the integral of f from a to b.

Worked examples

If f(x) = 2x on [0, 1] (and 0 elsewhere), find P(0.25 ≤ X ≤ 0.75).
  1. ∫₀.₂₅^0.75 2x dx = [x²]₀.₂₅^0.75 = 0.5625 - 0.0625 = 0.5

Answer: 0.5 (50%)

Practice this and 135 more formulas in the CalcRef workspace — quizzes, reference tables, a 16-category unit converter, and an expression evaluator.