Continuous Random Variable — Definition, Formula & Examples

A continuous random variable is a random variable that can take any value in an interval, such as a person's height or a wait time. It does not take only separate, countable values the way a die roll does.

A random variable $X$ is continuous if there is a probability density function $f$ with $f(x) \ge 0$ for all $x$ and $\int_{-\infty}^{\infty} f(x)\,dx = 1$ , such that $P(a \le X \le b) = \int_a^b f(x)\,dx$ for all $a \le b$ . It follows that $P(X = c) = 0$ for every single value $c$ .

Key Formula

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

Where:

$X$ = the continuous random variable
$f(x)$ = the probability density function (the height of the density curve at x)
$a, b$ = the endpoints of the interval, with a ≤ b

Worked Example

Problem: A bus arrives at a random time within a 20-minute window, so your wait time

X

(in minutes) is uniform on

[0, 20]

with density

f(x) = \frac{1}{20}

. Find the probability that you wait between 5 and 12 minutes.

Step 1: Check that the density is valid. The total area under the curve must equal 1. Here the region under the curve is a rectangle with width 20 and height 1/20.

20 \times \frac{1}{20} = 1

Step 2: The probability is the area under the curve from 5 to 12. That region is a rectangle with width 12 − 5 = 7 and height 1/20.

P(5 \le X \le 12) = 7 \times \frac{1}{20} = \frac{7}{20} = 0.35

Step 3: Because a single point has zero width, it adds no area. So whether you include the endpoints does not change the answer.

P(X = 12) = 0 \quad\Rightarrow\quad P(5 < X < 12) = P(5 \le X \le 12)

Answer: The probability of waiting between 5 and 12 minutes is

0.35

Why It Matters

In AP Statistics, every normal-model calculation treats the variable as continuous. When you convert to a z-score and read a table, you are finding an area under a density curve. The same idea gives p-values for t-tests and chi-square tests, and engineers use it to model component lifetimes with the exponential distribution.

Common Mistakes

Mistake: Treating the density value

f(x)

as the probability that

X = x

Correction: Only areas under the curve are probabilities, and

P(X = x) = 0

. A density can even exceed 1. For example, a uniform variable on

[0, 0.5]

has

f(x) = 2

, yet its total area is

0.5 \times 2 = 1

Related Terms

Random Variable — the broader concept this term builds on
Uniform Distribution — Simplest continuous distribution with flat density
Exponential Distribution — Continuous model for waiting times and lifetimes
Standard Normal Distribution Table — Gives areas under the standard normal curve
Chi-Squared Distribution — Continuous distribution used in chi-square tests
Beta Distribution — Continuous distribution on the interval from 0 to 1
Bernoulli Distribution — Discrete contrast with only two possible values