Continuous Probability Distributions — Mathematics Knowledge System

Overview

A continuous probability distribution describes a random variable that can take any value in an interval of real numbers, such as height, time, or temperature. Unlike discrete distributions, the probability of any single exact value is 0; instead, probabilities are computed as areas under a probability density function (PDF) f(x) over intervals. A valid PDF must be nonnegative everywhere and integrate to 1 over its entire support. Common continuous distributions include the uniform, normal (Gaussian), exponential, and gamma distributions, each arising from different physical or statistical modeling assumptions.

Intuition

Picture pouring a fixed amount of sand (representing total probability 1) along a number line so it forms a smooth pile — the height of the pile at any point is the density f(x). No single infinitesimally thin slice holds any sand by itself (probability of an exact point is 0), but the amount of sand between two markers a and b — the area under the curve — tells you the probability the random variable falls in that range. This is why continuous probability requires integration instead of summation.

Formal Definition

Definition

A probability density function f: ℝ → [0,∞) must satisfy:

f(x) \ge 0 \quad \text{for all } x

Nonnegativity

\int_{-\infty}^{\infty} f(x)\,dx = 1

Normalization

P(a \le X \le b) = \int_a^b f(x)\,dx

Probability as area under the curve

E[X] = \int_{-\infty}^{\infty} x\,f(x)\,dx

Expected value

Worked Examples

The probability is the area under the constant density between 1 and 3.
$P(1 \le X \le 3) = \int_1^3 \tfrac{1}{4}\,dx = \tfrac{1}{4}(3-1)$
Evaluate the width times the height.
$P(1 \le X \le 3) = \tfrac{1}{4}\cdot 2 = \tfrac{1}{2}$

Answer: P(1 ≤ X ≤ 3) = 1/2.

Practice Problems

Difficulty 3/10

X is uniform on [0,10]. Find P(2 ≤ X ≤ 6).

Difficulty 4/10

Find c so that f(x) = c for 0 ≤ x ≤ 5 (and 0 elsewhere) is a valid PDF.

Difficulty 5/10

A bus arrival time X (minutes after the hour) is uniform on [0,20]. What is the probability you wait 5 minutes or less if you arrive right at the hour (i.e., P(X ≤ 5))?

Quiz

Summary

A continuous distribution is described by a PDF f(x) ≥ 0 with total integral 1; individual points have probability 0.
Probabilities correspond to areas under the density curve: P(a≤X≤b) = ∫ₐᵇ f(x)dx.
Common examples include the uniform, normal, and exponential distributions.

References

WebsiteWikipedia — Probability density function