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.

  • 2 worked examples
  • 14 in Applications of Integrals

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%)

1 more worked examplePremium

A waiting time has density f(x) = 2e^(-2x) for x ≥ 0. Find P(X ≥ 1).

Step-by-step solution locked.

Unlock the full working. CalcRef Premium opens the step-by-step solution above, and 141 worked solutions across the whole library - the intermediate lines, not just the answer. The formula statement, its conditions, its variables and the first worked example stay free on every page.
See PremiumBrowse the free reference

See all 14 Applications of Integrals formulas on one printable sheet - the same statements and conditions as these pages, typeset and laid out for paper.

Common questions

What is the Probability (Continuous)?

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. It is one of the applications of integrals formulas in the CalcRef reference.

How do you use the Probability (Continuous)?

Worked example. If f(x) = 2x on [0, 1] (and 0 elsewhere), find P(0.25 ≤ X ≤ 0.75). ∫₀.₂₅^0.75 2x dx = [x²]₀.₂₅^0.75 = 0.5625 - 0.0625 = 0.5. Answer: 0.5 (50%).

Practice

Quiz yourself on Applications of Integrals

Probability (Continuous) is one of 14 Applications of Integrals formulas in CalcRef, out of 136 in the library. Today's free round is 10 mixed multiple-choice questions drawn from the same dataset as this page, refreshed every day and replayable as often as you like - no account needed.

Take today's free quizAll 14 Applications of Integrals formulas